a new one-sample test for goodness-of-fit
TRANSCRIPT
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A New One-Sample Test for Goodness-of-FitJoe Damico a ba Office of Information Technology The Ohio State University Columbus Ohio USAb Ohio State University Office of Information Technology 1121 Kinnear RoadColumbus OH 43212 USAPublished online 16 Aug 2006
To cite this article Joe Damico (2005) A New One-Sample Test for Goodness-of-Fit Communications in Statistics - Theoryand Methods 331 181-193 DOI 101081STA-120026585
To link to this article httpdxdoiorg101081STA-120026585
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A New One-Sample Test for Goodness-of-Fit
Joe Damico
Office of Information Technology The Ohio State UniversityColumbus Ohio USA
ABSTRACT
In this article we describe a new statistic It has a number of qualitiesthat recommend it for use as a one-sample test for goodness-of-fit
It is easy to describe and compute and so is useful as ateaching tool
It is a distribution-free statistic
Its distribution is skewed and it has a comparativelylarge range of values Therefore it can supply more criticalpoints that correspondto desired alpha levels
We can determine the 01 05 10 and 20 critical points forany large value of n by using a generalized formula
Correspondence Joe Damico Ohio State University Office of InformationTechnology 1121 Kinnear Road Columbus OH 43212 USA E-mail Damico1osuedu
181
DOI 101081STA-120026585 0361-0926 (Print) 1532-415X (Online)
Copyright 2004 by Marcel Dekker Inc wwwdekkercom
COMMUNICATIONS IN STATISTICS
Theory and Methods
Vol 33 No 1 pp 181ndash193 2004
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We can extend the definition of this statistic to a two-samplesituation
It is a test that provides excellent power My results show thatthe power is on a par with the Cramer-Von Mises one-sampletest for goodness-of-fit
This article contains five sections as follows
1 Defining the new statistic A
2 Description of the tests of power
3 Tables of the distribution for the A statistic
4 Table summarizing the results of the power tests
5 A brief bibliography
Key Words Distribution-free Goodness-of-fit Greatest integerfunction Non-parametric test One-sample test Two-sample test
1 DEFINITION OF THE A STATISTIC
Let Xi ifrac14 1 n denote n observations from a population PxThat is the Xi are independent and identically distributed We wish totest H0 Px is a population with some specified probability density func-tion (pdf )
Begin by partitioning the range of the pdf into n equal and non-over-lapping intervals (That is the integral of the pdf over each of these inter-valsfrac14 1=n) Under the null hypothesis we would expect one observationfrom our sample to occur in each of these n intervals (The expectedvalues of the order statistics are i=(nthorn 1) where i is an integer in therange from 1 to n inclusive) Our one-sample A statistic is a measureof how much the actual observations deviate from this expectation
If you think of this situation as analogous to a row of boxes each ofwhich contains from 0 to n balls with the total of all balls equal to thetotal number of boxes (n) then you may think of computing this statisticas equivalent to computing the minimum number of lsquolsquomovesrsquorsquo required toproduce a row of boxes with one and only one ball in each box And alsquolsquomoversquorsquo is defined as a move of one ball from one box to an adjacentbox An example may be useful
Suppose we have a random sample comprising the following sevenvalues 9088 9328 9351 1023 1044 1048 and 1053 We wish to testthe hypothesis that these seven values were drawn from a normal distri-bution with meanfrac14 100 and standard deviationfrac14 5 We compute the
182 Damico
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z-scores and the corresponding values from the normal distribution(1824) 0034 (1344) 0089 (1298) 0097 (046) 0678 (088)0811 (096) 0832 and (106) 0855 We proceed as if we were testingthe null hypothesis that these seven decimal numbers (0034 through0855) came from a uniform distribution on the interval (000 100)We begin by defining seven intervals and determining the number ofobservations in each
(0000 0143) 3(0143 0286) 0(0286 0429) 0(0429 0571) 0(0571 0714) 1(0714 0857) 3(0857 1000) 0
We now determine the number of moves required to produce the lsquolsquogroundstatersquorsquo (ie one ball in each box)
1stmove
2ndmove
3rdmove
4thmove
5thmove
6thmove
(0000 0143) 3 2 2 1 1 1 1(0143 0286) 0 1 0 1 1 1 1(0286 0429) 0 0 1 1 1 1 1(0429 0571) 0 0 0 0 0 1 1(0571 0714) 1 1 1 1 2 1 1(0714 0857) 3 3 3 3 2 2 1(0857 1000) 0 0 0 0 0 0 1
So the computed value of the A statistic is 6 The probability under thenull hypothesis that the A statistic assumes a value gtfrac14 6 is 03868 Thisalpha level would generally not be considered significant and so the nullhypothesis would not be rejected
The mathematical description of the statistic is simply a summationof a number of terms We begin with a few definitions As already men-tioned the Xi ifrac14 1 n denote n observations from a population PxLet Si ifrac14 1 n denote the order statistics Let F(si) be the value of thecumulative distribution function for the ith order statistic Finally letGif( ) represent the greatest integer function We define A as follows
A frac14Xn
ifrac141
jGifethnFethsiTHORN thorn 1THORN ij
A New One-Sample Test for Goodness-of-Fit 183
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So for our example
Gifeth7 0034thorn 1THORN frac14 1 j1 1j frac14 0
Gifeth7 0089thorn 1THORN frac14 1 j1 2j frac14 1
Gifeth7 0097thorn 1THORN frac14 1 j1 3j frac14 2
Gifeth7 0678thorn 1THORN frac14 5 j5 4j frac14 1
Gifeth7 0812thorn 1THORN frac14 6 j6 5j frac14 1
Gifeth7 0834thorn 1THORN frac14 6 j6 6j frac14 0
Gifeth7 0855thorn 1THORN frac14 6 j6 7j frac14 1
And the value of A is 0thorn 1thorn 2thorn 1thorn 1thorn 0thorn 1frac14 6 as before It should beobvious that the absolute values of the seven differences computed aboveactually correspond to the number of lsquolsquomovesrsquorsquo required to place each ofthe n order statistics into its corresponding lsquolsquoboxrsquorsquo Thus the third orderstatistic which lsquolsquooriginated in box 1rsquorsquo must be moved twice in order tolsquolsquoend up in box 3rsquorsquo
This statistic shares with the chi-square the concept of partitioningthe range of the probability distribution However it has the decidedadvantage of being useful in situations where we are able to obtain onlya small sample
In section three of this article I have published the exact distributionof the A statistic for values of nfrac14 2 3 4 5 6 and 7 For larger values of nI used computer simulations to generate the approximate distributions
After completing this work I was able to compute some formulaethat give excellent predictions of the critical values for large values of n(These formulae were computed by using stepwise regression techniques)
The resulting formulae (for alpha levelsfrac14 01 05 10 and 20) aregiven in the Table 1 The values computed from these formulae fornfrac14 20 30 40 50 60 70 80 90 and 100 are given in Table 2
Obviously these are good estimations of the values computedthrough the simulation and appearing in section three of this article
Another interesting fact is that we could easily extend the definitionof this statistic to test the two-sample problem in the following way
Let Xi ifrac14 1 m denote m observations from a population Pxand let F(x) be the cdf Let yj jfrac14 1 n denote n observations froma second population Py and let G(y) be the cdf We wish to test the nullhypothesis Ffrac14G against the broad alternative hypothesis F not frac14 GAssume that m is less than or equal to n Let tk kfrac14 1 (mthorn n)be the order statistics from the combined samples We construct thetwo-sample statistic A2 using the ranks (rirsquos) of the original xirsquos in the
184 Damico
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combined sample
A2 frac14Xm
ifrac141
jGifethethriTHORN ethm=ethmthorn nthorn 1THORNTHORN thorn 1THORN ij
Example We have a sample of size 4 from the x population and asample of size 6 from the y population
The x observations (and ranks) are
089eth1THORN 097eth2THORN 125eth3THORN 498eth5THORN
The y observations (and ranks) are
134eth4THORN 678eth6THORN 714eth7THORN 834eth8THORN 912eth9THORN 934eth10THORN
We compute
Gifetheth1THORN eth4=11THORN thorn 1THORN frac14 1 j1 1j frac14 0
Gifetheth2THORN eth4=11THORN thorn 1THORN frac14 1 j1 2j frac14 1
Gifetheth3THORN eth4=11THORN thorn 1THORN frac14 2 j2 3j frac14 1
Gifetheth5THORN eth4=11THORN thorn 1THORN frac14 2 j2 4j frac14 2
Table 2
Values of n
Alpha levels 20 30 40 50 60 70 80 90 100
020 37 68 104 145 190 240 293 349 409010 45 82 127 177 232 292 357 426 499005 52 96 147 205 270 340 415 495 580001 67 123 190 265 348 438 535 639 748
Table 1
Alpha level Critical value
020 Gif(04081 n(3=2)thorn 1)010 Gif(04981 n(3=2)thorn 1)005 Gif(05797 n(3=2)thorn 1)001 Gif(07473 n(3=2)thorn 1)
A New One-Sample Test for Goodness-of-Fit 185
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So the statistic A2 has the value 4 The probability under the nullhypothesis that the A2 statistic assumes a value gtfrac14 4 is 00286 So thenull hypothesis would be rejected The cumulative distribution functionfor this two-sample test statistic (with mfrac14 4 and nfrac14 6) looks like this
2 POWER OF THE A STATISTIC
In an article titled lsquolsquoEDF Statistics for Goodness-of-fit and SomeComparisonsrsquorsquo Stephens 1974 presented a chart comparing the powersof several different statistical tests The null hypothesis is that we havea uniform random number on the interval (0 1) There were sevenalternate distributions defined as follows
F FethxTHORN frac14 1 eth1 zTHORNk 0 ltfrac14 z ltfrac14 1 k frac14 15 2
G FethxTHORN frac14 2ethk1THORNzk 0 ltfrac14 z ltfrac14 5
FethxTHORN frac14 1 2ethk1THORNeth1 zTHORNk 5 ltfrac14 z ltfrac14 1 k frac14 15 2 3
H FethxTHORN frac14 05 2ethk1THORNeth5 zTHORNk 0 ltfrac14 z ltfrac14 5
FethxTHORN frac14 05thorn 2ethk1THORNethz 5THORNk 5 ltfrac14 z ltfrac14 1 k frac14 15 2
I computed the power of the A statistic in these same seven situationsThe results are shown in section four of this paper The power of theA statistic compares very favorably with both the Kolmogorov-SmirnovD statistic and the Cramer-von Mises W2 statistic
3 TABLES OF THE A STATISTIC
Tables of the exact distribution of A for sample sizes (n) rangingfrom 2 through 7 is shown in Table 3
The tables of the A statistic (Table 4) were computed using simula-tions The number of replications (N) is shown followed by the samplesize (n) and four critical values with the corresponding alpha levels
Value 0 1 2 3 4Cumulative density 1714 5143 8381 9714 1000
186 Damico
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In each case the four alpha levels were selected to be near 020 010 005and 001
Here are some additional critical values for values of n from 8through 15 (Tables 5 and 6)
4 TABLES COMPARING POWER OF THE
A STATISTIC WITH OTHER
ONE-SAMPLE TESTS
As noted earlier the table of results comparing Kolmogorov-Smirnov D Cramer-von Mises W2 Kuiper V Watson U2 Anderson-Darling A2 Q (frac14Si ln zi) and Chi2 first appeared in an article byStephens (1974) in JASA I have simply added the column showing thepower of the A statistic as estimated by a number of simulations
Table 3
A nfrac14 2 nfrac14 3 nfrac14 4 nfrac14 5 nfrac14 6 nfrac14 7
0 0500 022222 009375 003840 001543 000611991 1000 066667 037500 019200 009259 004283932 092593 068750 044160 025720 013973773 100000 087500 067200 046296 029273524 096094 082560 064472 046256235 099219 091520 077761 061317996 100000 096352 086703 073131097 098656 092438 081845158 099616 095945 088053459 099936 097990 0923662710 100000 099096 0952885311 099640 0972077912 099880 0984233913 099970 0991597214 099996 0995813215 100000 0998074216 0999198617 0999708518 0999912619 0999980620 0999997621 10000000
A New One-Sample Test for Goodness-of-Fit 187
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Table 4
N nCritical
value (A) P(A) gtfrac14A N nCritical
value (A) P(A) gtfrac14A
10000 8 9 02267 1000 18 32 019611 01183 38 010113 00543 44 004917 00089 57 0009
10000 9 11 02074 1000 19 33 020413 01176 40 010216 00462 48 004920 00106 63 0010
10000 10 13 02008 10000 20 37 0197316 00972 45 0099319 00406 53 0047724 00090 66 00092
10000 11 15 02018 1000 21 40 019719 00860 49 009521 00522 56 005127 00111 75 0009
10000 12 17 02015 1000 22 43 020020 01162 51 010024 00507 59 004930 00099 77 0010
10000 13 19 02056 1000 23 47 019823 01038 56 010027 00484 66 004935 00097 83 0010
10000 14 22 01909 1200 24 48 0200827 00945 58 0098331 00497 67 0049240 00085 87 00092
10000 15 23 02125 10000 25 51 0199129 00972 61 0103134 00468 72 0051543 00089 93 00107
1000 16 26 0192 1600 26 53 0203131 0092 65 0101336 0046 77 0048148 0010 97 00094
1000 17 29 0193 1600 27 57 0196934 0102 69 0098840 0045 80 0047551 0008 104 00100
188 Damico
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Table 4 Continued
N nCritical
value (A) P(A) gtfrac14A N nCritical
value (A) P(A) gtfrac14A
1600 28 59 01981 1600 38 96 02019
73 01019 118 0101386 00500 135 00500110 00100 181 00100
1600 29 66 01975 1600 39 99 0201380 01013 120 0100092 00481 140 00494117 00100 170 00094
10000 30 67 02029 10000 40 103 0201181 01025 126 0100095 00490 148 00496126 00095 191 00099
1600 31 71 01969 10000 50 145 0194587 00981 177 00984101 00500 205 00496128 00100 265 00099
1600 32 74 02038 10000 60 190 0199991 01000 232 00986107 00488 270 00498135 00100 348 00093
1600 33 77 01963 10000 70 240 0201990 01006 292 01027108 00500 340 00488141 00094 438 00095
1600 34 82 01975 10000 80 293 01988101 00988 357 01015117 00488 415 00530149 00100 535 00112
10000 35 84 02036 10000 90 349 01982102 01019 426 01015120 00493 495 00530152 00103 639 00112
1600 36 91 02019 10000 100 409 01930113 00981 499 00955132 00488 580 00467166 00100 748 00089
1600 37 90 01988110 01000128 00488172 00100
A New One-Sample Test for Goodness-of-Fit 189
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Table
5
nfrac148
nfrac149
nfrac1410
nfrac1411
Critical
value(A
)P(A
)gtfrac14A
Critical
value(A
)P(A
)gtfrac14A
Critical
value(A
)P(A
)gtfrac14A
Critical
value(A
)P(A
)gtfrac14A
902267
11
02074
13
02008
15
02018
10
01657
12
01580
14
01606
16
01628
11
01183
13
01176
15
01260
17
01336
12
00810
14
00874
16
00972
18
01091
13
00543
15
00633
17
00729
19
00860
14
00347
16
00462
18
00542
20
00670
15
00227
17
00335
19
00406
21
00522
16
00151
18
00241
20
00298
22
00417
17
00089
19
00157
21
00221
23
00334
20
00106
22
00166
24
00257
21
00062
23
00126
25
00194
24
00090
26
00147
27
00111
28
00082
190 Damico
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Table
6
nfrac1412
nfrac1413
nfrac1414
nfrac1415
Critical
value(A
)P(A
)gtfrac14A
Critical
value(A
)P(A
)gtfrac14A
Critical
value(A
)P(A
)gtfrac14A
Critical
value(A
)P(A
)gtfrac14A
17
02015
19
02056
21
02168
23
02125
18
01677
20
01733
22
01909
24
01873
19
01407
21
01488
23
01657
25
01648
20
01162
22
01249
24
01431
26
01454
21
00929
23
01038
25
01267
27
01285
22
00755
24
00878
26
01100
28
01132
23
00615
25
00736
27
00945
29
00972
24
00507
26
00606
28
00806
30
00841
25
00416
27
00484
29
00691
31
00723
26
00315
28
00410
30
00581
32
00624
27
00240
29
00334
31
00497
33
00542
28
00175
30
00278
32
00435
34
00468
29
00134
31
00221
33
00360
35
00403
30
00099
32
00184
34
00293
36
00344
33
00145
35
00236
37
00296
34
00120
36
00200
38
00240
35
00097
37
00175
39
00206
38
00141
40
00163
39
00119
41
00141
40
00085
42
00118
43
00089
A New One-Sample Test for Goodness-of-Fit 191
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(Table 7) The alternative distribution functions are described in sectiontwo of this paper The sample sizes (n) were set equal to 10 20 and 40The alpha level was set at 010
REFERENCES
Bradley J V (1968) Distribution-Free Statistical Tests EnglewoodCliffs NJ Prentice-Hall
Gibbons J D (1976) Nonparametric Methods for Quantitative AnalysisNew York Holt Rinehart and Winston
Gibbons J D (1985) Nonparametric Statistical Inference New YorkMarcel Dekker
Table 7
Alternative D W2 V U2 A2 Q Chi2 A
(nfrac14 10 alphafrac14 010)Fkfrac1415 023 027 018 019 024 043 ndash 027Fkfrac1420 054 060 035 035 058 ndash ndash 058Gkfrac1415 009 007 022 023 006 ndash ndash 007Gkfrac1420 009 007 040 044 006 ndash ndash 010Gkfrac1430 021 021 081 086 018 ndash ndash 029Hkfrac1415 ndash ndash ndash ndash ndash ndash ndash 014Hkfrac1420 ndash ndash ndash ndash ndash ndash ndash 021
(nfrac14 20 alphafrac14 010)Fkfrac1415 038 046 025 028 046 068 ndash 046Fkfrac1420 078 087 061 060 087 097 059 088Gkfrac1415 013 011 032 034 010 011 ndash 010Gkfrac1420 025 025 071 077 028 025 ndash 029Gkfrac1430 063 079 099 099 084 ndash ndash 083Hkfrac1415 025 020 036 037 028 ndash ndash 017Hkfrac1420 047 044 071 077 054 ndash ndash 036
(nfrac14 40 alphafrac14 010)Fkfrac1415 060 070 043 043 ndash 089 040 073Fkfrac1420 098 099 091 089 ndash 100 089 099Gkfrac1415 019 022 057 061 ndash ndash 039 023Gkfrac1420 056 072 096 098 ndash ndash 085 075Gkfrac1430 ndash ndash ndash ndash ndash ndash ndash 100Hkfrac1415 036 032 058 063 ndash ndash ndash 027Hkfrac1420 071 080 096 098 ndash ndash ndash 075
192 Damico
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Hollander M Wolfe D A (1973) Nonparametric Statistical Methods2nd ed New York Wiley
Stephens M A (1974) EDF statistics for goodness-of-fit and somecomparisons (In theory and methods) J Amer Stat Assoc Theoryand Methods Section September 69(347)730ndash737
A New One-Sample Test for Goodness-of-Fit 193
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A New One-Sample Test for Goodness-of-Fit
Joe Damico
Office of Information Technology The Ohio State UniversityColumbus Ohio USA
ABSTRACT
In this article we describe a new statistic It has a number of qualitiesthat recommend it for use as a one-sample test for goodness-of-fit
It is easy to describe and compute and so is useful as ateaching tool
It is a distribution-free statistic
Its distribution is skewed and it has a comparativelylarge range of values Therefore it can supply more criticalpoints that correspondto desired alpha levels
We can determine the 01 05 10 and 20 critical points forany large value of n by using a generalized formula
Correspondence Joe Damico Ohio State University Office of InformationTechnology 1121 Kinnear Road Columbus OH 43212 USA E-mail Damico1osuedu
181
DOI 101081STA-120026585 0361-0926 (Print) 1532-415X (Online)
Copyright 2004 by Marcel Dekker Inc wwwdekkercom
COMMUNICATIONS IN STATISTICS
Theory and Methods
Vol 33 No 1 pp 181ndash193 2004
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We can extend the definition of this statistic to a two-samplesituation
It is a test that provides excellent power My results show thatthe power is on a par with the Cramer-Von Mises one-sampletest for goodness-of-fit
This article contains five sections as follows
1 Defining the new statistic A
2 Description of the tests of power
3 Tables of the distribution for the A statistic
4 Table summarizing the results of the power tests
5 A brief bibliography
Key Words Distribution-free Goodness-of-fit Greatest integerfunction Non-parametric test One-sample test Two-sample test
1 DEFINITION OF THE A STATISTIC
Let Xi ifrac14 1 n denote n observations from a population PxThat is the Xi are independent and identically distributed We wish totest H0 Px is a population with some specified probability density func-tion (pdf )
Begin by partitioning the range of the pdf into n equal and non-over-lapping intervals (That is the integral of the pdf over each of these inter-valsfrac14 1=n) Under the null hypothesis we would expect one observationfrom our sample to occur in each of these n intervals (The expectedvalues of the order statistics are i=(nthorn 1) where i is an integer in therange from 1 to n inclusive) Our one-sample A statistic is a measureof how much the actual observations deviate from this expectation
If you think of this situation as analogous to a row of boxes each ofwhich contains from 0 to n balls with the total of all balls equal to thetotal number of boxes (n) then you may think of computing this statisticas equivalent to computing the minimum number of lsquolsquomovesrsquorsquo required toproduce a row of boxes with one and only one ball in each box And alsquolsquomoversquorsquo is defined as a move of one ball from one box to an adjacentbox An example may be useful
Suppose we have a random sample comprising the following sevenvalues 9088 9328 9351 1023 1044 1048 and 1053 We wish to testthe hypothesis that these seven values were drawn from a normal distri-bution with meanfrac14 100 and standard deviationfrac14 5 We compute the
182 Damico
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ORDER REPRINTS
z-scores and the corresponding values from the normal distribution(1824) 0034 (1344) 0089 (1298) 0097 (046) 0678 (088)0811 (096) 0832 and (106) 0855 We proceed as if we were testingthe null hypothesis that these seven decimal numbers (0034 through0855) came from a uniform distribution on the interval (000 100)We begin by defining seven intervals and determining the number ofobservations in each
(0000 0143) 3(0143 0286) 0(0286 0429) 0(0429 0571) 0(0571 0714) 1(0714 0857) 3(0857 1000) 0
We now determine the number of moves required to produce the lsquolsquogroundstatersquorsquo (ie one ball in each box)
1stmove
2ndmove
3rdmove
4thmove
5thmove
6thmove
(0000 0143) 3 2 2 1 1 1 1(0143 0286) 0 1 0 1 1 1 1(0286 0429) 0 0 1 1 1 1 1(0429 0571) 0 0 0 0 0 1 1(0571 0714) 1 1 1 1 2 1 1(0714 0857) 3 3 3 3 2 2 1(0857 1000) 0 0 0 0 0 0 1
So the computed value of the A statistic is 6 The probability under thenull hypothesis that the A statistic assumes a value gtfrac14 6 is 03868 Thisalpha level would generally not be considered significant and so the nullhypothesis would not be rejected
The mathematical description of the statistic is simply a summationof a number of terms We begin with a few definitions As already men-tioned the Xi ifrac14 1 n denote n observations from a population PxLet Si ifrac14 1 n denote the order statistics Let F(si) be the value of thecumulative distribution function for the ith order statistic Finally letGif( ) represent the greatest integer function We define A as follows
A frac14Xn
ifrac141
jGifethnFethsiTHORN thorn 1THORN ij
A New One-Sample Test for Goodness-of-Fit 183
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So for our example
Gifeth7 0034thorn 1THORN frac14 1 j1 1j frac14 0
Gifeth7 0089thorn 1THORN frac14 1 j1 2j frac14 1
Gifeth7 0097thorn 1THORN frac14 1 j1 3j frac14 2
Gifeth7 0678thorn 1THORN frac14 5 j5 4j frac14 1
Gifeth7 0812thorn 1THORN frac14 6 j6 5j frac14 1
Gifeth7 0834thorn 1THORN frac14 6 j6 6j frac14 0
Gifeth7 0855thorn 1THORN frac14 6 j6 7j frac14 1
And the value of A is 0thorn 1thorn 2thorn 1thorn 1thorn 0thorn 1frac14 6 as before It should beobvious that the absolute values of the seven differences computed aboveactually correspond to the number of lsquolsquomovesrsquorsquo required to place each ofthe n order statistics into its corresponding lsquolsquoboxrsquorsquo Thus the third orderstatistic which lsquolsquooriginated in box 1rsquorsquo must be moved twice in order tolsquolsquoend up in box 3rsquorsquo
This statistic shares with the chi-square the concept of partitioningthe range of the probability distribution However it has the decidedadvantage of being useful in situations where we are able to obtain onlya small sample
In section three of this article I have published the exact distributionof the A statistic for values of nfrac14 2 3 4 5 6 and 7 For larger values of nI used computer simulations to generate the approximate distributions
After completing this work I was able to compute some formulaethat give excellent predictions of the critical values for large values of n(These formulae were computed by using stepwise regression techniques)
The resulting formulae (for alpha levelsfrac14 01 05 10 and 20) aregiven in the Table 1 The values computed from these formulae fornfrac14 20 30 40 50 60 70 80 90 and 100 are given in Table 2
Obviously these are good estimations of the values computedthrough the simulation and appearing in section three of this article
Another interesting fact is that we could easily extend the definitionof this statistic to test the two-sample problem in the following way
Let Xi ifrac14 1 m denote m observations from a population Pxand let F(x) be the cdf Let yj jfrac14 1 n denote n observations froma second population Py and let G(y) be the cdf We wish to test the nullhypothesis Ffrac14G against the broad alternative hypothesis F not frac14 GAssume that m is less than or equal to n Let tk kfrac14 1 (mthorn n)be the order statistics from the combined samples We construct thetwo-sample statistic A2 using the ranks (rirsquos) of the original xirsquos in the
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combined sample
A2 frac14Xm
ifrac141
jGifethethriTHORN ethm=ethmthorn nthorn 1THORNTHORN thorn 1THORN ij
Example We have a sample of size 4 from the x population and asample of size 6 from the y population
The x observations (and ranks) are
089eth1THORN 097eth2THORN 125eth3THORN 498eth5THORN
The y observations (and ranks) are
134eth4THORN 678eth6THORN 714eth7THORN 834eth8THORN 912eth9THORN 934eth10THORN
We compute
Gifetheth1THORN eth4=11THORN thorn 1THORN frac14 1 j1 1j frac14 0
Gifetheth2THORN eth4=11THORN thorn 1THORN frac14 1 j1 2j frac14 1
Gifetheth3THORN eth4=11THORN thorn 1THORN frac14 2 j2 3j frac14 1
Gifetheth5THORN eth4=11THORN thorn 1THORN frac14 2 j2 4j frac14 2
Table 2
Values of n
Alpha levels 20 30 40 50 60 70 80 90 100
020 37 68 104 145 190 240 293 349 409010 45 82 127 177 232 292 357 426 499005 52 96 147 205 270 340 415 495 580001 67 123 190 265 348 438 535 639 748
Table 1
Alpha level Critical value
020 Gif(04081 n(3=2)thorn 1)010 Gif(04981 n(3=2)thorn 1)005 Gif(05797 n(3=2)thorn 1)001 Gif(07473 n(3=2)thorn 1)
A New One-Sample Test for Goodness-of-Fit 185
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ORDER REPRINTS
So the statistic A2 has the value 4 The probability under the nullhypothesis that the A2 statistic assumes a value gtfrac14 4 is 00286 So thenull hypothesis would be rejected The cumulative distribution functionfor this two-sample test statistic (with mfrac14 4 and nfrac14 6) looks like this
2 POWER OF THE A STATISTIC
In an article titled lsquolsquoEDF Statistics for Goodness-of-fit and SomeComparisonsrsquorsquo Stephens 1974 presented a chart comparing the powersof several different statistical tests The null hypothesis is that we havea uniform random number on the interval (0 1) There were sevenalternate distributions defined as follows
F FethxTHORN frac14 1 eth1 zTHORNk 0 ltfrac14 z ltfrac14 1 k frac14 15 2
G FethxTHORN frac14 2ethk1THORNzk 0 ltfrac14 z ltfrac14 5
FethxTHORN frac14 1 2ethk1THORNeth1 zTHORNk 5 ltfrac14 z ltfrac14 1 k frac14 15 2 3
H FethxTHORN frac14 05 2ethk1THORNeth5 zTHORNk 0 ltfrac14 z ltfrac14 5
FethxTHORN frac14 05thorn 2ethk1THORNethz 5THORNk 5 ltfrac14 z ltfrac14 1 k frac14 15 2
I computed the power of the A statistic in these same seven situationsThe results are shown in section four of this paper The power of theA statistic compares very favorably with both the Kolmogorov-SmirnovD statistic and the Cramer-von Mises W2 statistic
3 TABLES OF THE A STATISTIC
Tables of the exact distribution of A for sample sizes (n) rangingfrom 2 through 7 is shown in Table 3
The tables of the A statistic (Table 4) were computed using simula-tions The number of replications (N) is shown followed by the samplesize (n) and four critical values with the corresponding alpha levels
Value 0 1 2 3 4Cumulative density 1714 5143 8381 9714 1000
186 Damico
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In each case the four alpha levels were selected to be near 020 010 005and 001
Here are some additional critical values for values of n from 8through 15 (Tables 5 and 6)
4 TABLES COMPARING POWER OF THE
A STATISTIC WITH OTHER
ONE-SAMPLE TESTS
As noted earlier the table of results comparing Kolmogorov-Smirnov D Cramer-von Mises W2 Kuiper V Watson U2 Anderson-Darling A2 Q (frac14Si ln zi) and Chi2 first appeared in an article byStephens (1974) in JASA I have simply added the column showing thepower of the A statistic as estimated by a number of simulations
Table 3
A nfrac14 2 nfrac14 3 nfrac14 4 nfrac14 5 nfrac14 6 nfrac14 7
0 0500 022222 009375 003840 001543 000611991 1000 066667 037500 019200 009259 004283932 092593 068750 044160 025720 013973773 100000 087500 067200 046296 029273524 096094 082560 064472 046256235 099219 091520 077761 061317996 100000 096352 086703 073131097 098656 092438 081845158 099616 095945 088053459 099936 097990 0923662710 100000 099096 0952885311 099640 0972077912 099880 0984233913 099970 0991597214 099996 0995813215 100000 0998074216 0999198617 0999708518 0999912619 0999980620 0999997621 10000000
A New One-Sample Test for Goodness-of-Fit 187
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Table 4
N nCritical
value (A) P(A) gtfrac14A N nCritical
value (A) P(A) gtfrac14A
10000 8 9 02267 1000 18 32 019611 01183 38 010113 00543 44 004917 00089 57 0009
10000 9 11 02074 1000 19 33 020413 01176 40 010216 00462 48 004920 00106 63 0010
10000 10 13 02008 10000 20 37 0197316 00972 45 0099319 00406 53 0047724 00090 66 00092
10000 11 15 02018 1000 21 40 019719 00860 49 009521 00522 56 005127 00111 75 0009
10000 12 17 02015 1000 22 43 020020 01162 51 010024 00507 59 004930 00099 77 0010
10000 13 19 02056 1000 23 47 019823 01038 56 010027 00484 66 004935 00097 83 0010
10000 14 22 01909 1200 24 48 0200827 00945 58 0098331 00497 67 0049240 00085 87 00092
10000 15 23 02125 10000 25 51 0199129 00972 61 0103134 00468 72 0051543 00089 93 00107
1000 16 26 0192 1600 26 53 0203131 0092 65 0101336 0046 77 0048148 0010 97 00094
1000 17 29 0193 1600 27 57 0196934 0102 69 0098840 0045 80 0047551 0008 104 00100
188 Damico
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Table 4 Continued
N nCritical
value (A) P(A) gtfrac14A N nCritical
value (A) P(A) gtfrac14A
1600 28 59 01981 1600 38 96 02019
73 01019 118 0101386 00500 135 00500110 00100 181 00100
1600 29 66 01975 1600 39 99 0201380 01013 120 0100092 00481 140 00494117 00100 170 00094
10000 30 67 02029 10000 40 103 0201181 01025 126 0100095 00490 148 00496126 00095 191 00099
1600 31 71 01969 10000 50 145 0194587 00981 177 00984101 00500 205 00496128 00100 265 00099
1600 32 74 02038 10000 60 190 0199991 01000 232 00986107 00488 270 00498135 00100 348 00093
1600 33 77 01963 10000 70 240 0201990 01006 292 01027108 00500 340 00488141 00094 438 00095
1600 34 82 01975 10000 80 293 01988101 00988 357 01015117 00488 415 00530149 00100 535 00112
10000 35 84 02036 10000 90 349 01982102 01019 426 01015120 00493 495 00530152 00103 639 00112
1600 36 91 02019 10000 100 409 01930113 00981 499 00955132 00488 580 00467166 00100 748 00089
1600 37 90 01988110 01000128 00488172 00100
A New One-Sample Test for Goodness-of-Fit 189
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Table
5
nfrac148
nfrac149
nfrac1410
nfrac1411
Critical
value(A
)P(A
)gtfrac14A
Critical
value(A
)P(A
)gtfrac14A
Critical
value(A
)P(A
)gtfrac14A
Critical
value(A
)P(A
)gtfrac14A
902267
11
02074
13
02008
15
02018
10
01657
12
01580
14
01606
16
01628
11
01183
13
01176
15
01260
17
01336
12
00810
14
00874
16
00972
18
01091
13
00543
15
00633
17
00729
19
00860
14
00347
16
00462
18
00542
20
00670
15
00227
17
00335
19
00406
21
00522
16
00151
18
00241
20
00298
22
00417
17
00089
19
00157
21
00221
23
00334
20
00106
22
00166
24
00257
21
00062
23
00126
25
00194
24
00090
26
00147
27
00111
28
00082
190 Damico
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Table
6
nfrac1412
nfrac1413
nfrac1414
nfrac1415
Critical
value(A
)P(A
)gtfrac14A
Critical
value(A
)P(A
)gtfrac14A
Critical
value(A
)P(A
)gtfrac14A
Critical
value(A
)P(A
)gtfrac14A
17
02015
19
02056
21
02168
23
02125
18
01677
20
01733
22
01909
24
01873
19
01407
21
01488
23
01657
25
01648
20
01162
22
01249
24
01431
26
01454
21
00929
23
01038
25
01267
27
01285
22
00755
24
00878
26
01100
28
01132
23
00615
25
00736
27
00945
29
00972
24
00507
26
00606
28
00806
30
00841
25
00416
27
00484
29
00691
31
00723
26
00315
28
00410
30
00581
32
00624
27
00240
29
00334
31
00497
33
00542
28
00175
30
00278
32
00435
34
00468
29
00134
31
00221
33
00360
35
00403
30
00099
32
00184
34
00293
36
00344
33
00145
35
00236
37
00296
34
00120
36
00200
38
00240
35
00097
37
00175
39
00206
38
00141
40
00163
39
00119
41
00141
40
00085
42
00118
43
00089
A New One-Sample Test for Goodness-of-Fit 191
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ORDER REPRINTS
(Table 7) The alternative distribution functions are described in sectiontwo of this paper The sample sizes (n) were set equal to 10 20 and 40The alpha level was set at 010
REFERENCES
Bradley J V (1968) Distribution-Free Statistical Tests EnglewoodCliffs NJ Prentice-Hall
Gibbons J D (1976) Nonparametric Methods for Quantitative AnalysisNew York Holt Rinehart and Winston
Gibbons J D (1985) Nonparametric Statistical Inference New YorkMarcel Dekker
Table 7
Alternative D W2 V U2 A2 Q Chi2 A
(nfrac14 10 alphafrac14 010)Fkfrac1415 023 027 018 019 024 043 ndash 027Fkfrac1420 054 060 035 035 058 ndash ndash 058Gkfrac1415 009 007 022 023 006 ndash ndash 007Gkfrac1420 009 007 040 044 006 ndash ndash 010Gkfrac1430 021 021 081 086 018 ndash ndash 029Hkfrac1415 ndash ndash ndash ndash ndash ndash ndash 014Hkfrac1420 ndash ndash ndash ndash ndash ndash ndash 021
(nfrac14 20 alphafrac14 010)Fkfrac1415 038 046 025 028 046 068 ndash 046Fkfrac1420 078 087 061 060 087 097 059 088Gkfrac1415 013 011 032 034 010 011 ndash 010Gkfrac1420 025 025 071 077 028 025 ndash 029Gkfrac1430 063 079 099 099 084 ndash ndash 083Hkfrac1415 025 020 036 037 028 ndash ndash 017Hkfrac1420 047 044 071 077 054 ndash ndash 036
(nfrac14 40 alphafrac14 010)Fkfrac1415 060 070 043 043 ndash 089 040 073Fkfrac1420 098 099 091 089 ndash 100 089 099Gkfrac1415 019 022 057 061 ndash ndash 039 023Gkfrac1420 056 072 096 098 ndash ndash 085 075Gkfrac1430 ndash ndash ndash ndash ndash ndash ndash 100Hkfrac1415 036 032 058 063 ndash ndash ndash 027Hkfrac1420 071 080 096 098 ndash ndash ndash 075
192 Damico
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Hollander M Wolfe D A (1973) Nonparametric Statistical Methods2nd ed New York Wiley
Stephens M A (1974) EDF statistics for goodness-of-fit and somecomparisons (In theory and methods) J Amer Stat Assoc Theoryand Methods Section September 69(347)730ndash737
A New One-Sample Test for Goodness-of-Fit 193
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We can extend the definition of this statistic to a two-samplesituation
It is a test that provides excellent power My results show thatthe power is on a par with the Cramer-Von Mises one-sampletest for goodness-of-fit
This article contains five sections as follows
1 Defining the new statistic A
2 Description of the tests of power
3 Tables of the distribution for the A statistic
4 Table summarizing the results of the power tests
5 A brief bibliography
Key Words Distribution-free Goodness-of-fit Greatest integerfunction Non-parametric test One-sample test Two-sample test
1 DEFINITION OF THE A STATISTIC
Let Xi ifrac14 1 n denote n observations from a population PxThat is the Xi are independent and identically distributed We wish totest H0 Px is a population with some specified probability density func-tion (pdf )
Begin by partitioning the range of the pdf into n equal and non-over-lapping intervals (That is the integral of the pdf over each of these inter-valsfrac14 1=n) Under the null hypothesis we would expect one observationfrom our sample to occur in each of these n intervals (The expectedvalues of the order statistics are i=(nthorn 1) where i is an integer in therange from 1 to n inclusive) Our one-sample A statistic is a measureof how much the actual observations deviate from this expectation
If you think of this situation as analogous to a row of boxes each ofwhich contains from 0 to n balls with the total of all balls equal to thetotal number of boxes (n) then you may think of computing this statisticas equivalent to computing the minimum number of lsquolsquomovesrsquorsquo required toproduce a row of boxes with one and only one ball in each box And alsquolsquomoversquorsquo is defined as a move of one ball from one box to an adjacentbox An example may be useful
Suppose we have a random sample comprising the following sevenvalues 9088 9328 9351 1023 1044 1048 and 1053 We wish to testthe hypothesis that these seven values were drawn from a normal distri-bution with meanfrac14 100 and standard deviationfrac14 5 We compute the
182 Damico
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z-scores and the corresponding values from the normal distribution(1824) 0034 (1344) 0089 (1298) 0097 (046) 0678 (088)0811 (096) 0832 and (106) 0855 We proceed as if we were testingthe null hypothesis that these seven decimal numbers (0034 through0855) came from a uniform distribution on the interval (000 100)We begin by defining seven intervals and determining the number ofobservations in each
(0000 0143) 3(0143 0286) 0(0286 0429) 0(0429 0571) 0(0571 0714) 1(0714 0857) 3(0857 1000) 0
We now determine the number of moves required to produce the lsquolsquogroundstatersquorsquo (ie one ball in each box)
1stmove
2ndmove
3rdmove
4thmove
5thmove
6thmove
(0000 0143) 3 2 2 1 1 1 1(0143 0286) 0 1 0 1 1 1 1(0286 0429) 0 0 1 1 1 1 1(0429 0571) 0 0 0 0 0 1 1(0571 0714) 1 1 1 1 2 1 1(0714 0857) 3 3 3 3 2 2 1(0857 1000) 0 0 0 0 0 0 1
So the computed value of the A statistic is 6 The probability under thenull hypothesis that the A statistic assumes a value gtfrac14 6 is 03868 Thisalpha level would generally not be considered significant and so the nullhypothesis would not be rejected
The mathematical description of the statistic is simply a summationof a number of terms We begin with a few definitions As already men-tioned the Xi ifrac14 1 n denote n observations from a population PxLet Si ifrac14 1 n denote the order statistics Let F(si) be the value of thecumulative distribution function for the ith order statistic Finally letGif( ) represent the greatest integer function We define A as follows
A frac14Xn
ifrac141
jGifethnFethsiTHORN thorn 1THORN ij
A New One-Sample Test for Goodness-of-Fit 183
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So for our example
Gifeth7 0034thorn 1THORN frac14 1 j1 1j frac14 0
Gifeth7 0089thorn 1THORN frac14 1 j1 2j frac14 1
Gifeth7 0097thorn 1THORN frac14 1 j1 3j frac14 2
Gifeth7 0678thorn 1THORN frac14 5 j5 4j frac14 1
Gifeth7 0812thorn 1THORN frac14 6 j6 5j frac14 1
Gifeth7 0834thorn 1THORN frac14 6 j6 6j frac14 0
Gifeth7 0855thorn 1THORN frac14 6 j6 7j frac14 1
And the value of A is 0thorn 1thorn 2thorn 1thorn 1thorn 0thorn 1frac14 6 as before It should beobvious that the absolute values of the seven differences computed aboveactually correspond to the number of lsquolsquomovesrsquorsquo required to place each ofthe n order statistics into its corresponding lsquolsquoboxrsquorsquo Thus the third orderstatistic which lsquolsquooriginated in box 1rsquorsquo must be moved twice in order tolsquolsquoend up in box 3rsquorsquo
This statistic shares with the chi-square the concept of partitioningthe range of the probability distribution However it has the decidedadvantage of being useful in situations where we are able to obtain onlya small sample
In section three of this article I have published the exact distributionof the A statistic for values of nfrac14 2 3 4 5 6 and 7 For larger values of nI used computer simulations to generate the approximate distributions
After completing this work I was able to compute some formulaethat give excellent predictions of the critical values for large values of n(These formulae were computed by using stepwise regression techniques)
The resulting formulae (for alpha levelsfrac14 01 05 10 and 20) aregiven in the Table 1 The values computed from these formulae fornfrac14 20 30 40 50 60 70 80 90 and 100 are given in Table 2
Obviously these are good estimations of the values computedthrough the simulation and appearing in section three of this article
Another interesting fact is that we could easily extend the definitionof this statistic to test the two-sample problem in the following way
Let Xi ifrac14 1 m denote m observations from a population Pxand let F(x) be the cdf Let yj jfrac14 1 n denote n observations froma second population Py and let G(y) be the cdf We wish to test the nullhypothesis Ffrac14G against the broad alternative hypothesis F not frac14 GAssume that m is less than or equal to n Let tk kfrac14 1 (mthorn n)be the order statistics from the combined samples We construct thetwo-sample statistic A2 using the ranks (rirsquos) of the original xirsquos in the
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combined sample
A2 frac14Xm
ifrac141
jGifethethriTHORN ethm=ethmthorn nthorn 1THORNTHORN thorn 1THORN ij
Example We have a sample of size 4 from the x population and asample of size 6 from the y population
The x observations (and ranks) are
089eth1THORN 097eth2THORN 125eth3THORN 498eth5THORN
The y observations (and ranks) are
134eth4THORN 678eth6THORN 714eth7THORN 834eth8THORN 912eth9THORN 934eth10THORN
We compute
Gifetheth1THORN eth4=11THORN thorn 1THORN frac14 1 j1 1j frac14 0
Gifetheth2THORN eth4=11THORN thorn 1THORN frac14 1 j1 2j frac14 1
Gifetheth3THORN eth4=11THORN thorn 1THORN frac14 2 j2 3j frac14 1
Gifetheth5THORN eth4=11THORN thorn 1THORN frac14 2 j2 4j frac14 2
Table 2
Values of n
Alpha levels 20 30 40 50 60 70 80 90 100
020 37 68 104 145 190 240 293 349 409010 45 82 127 177 232 292 357 426 499005 52 96 147 205 270 340 415 495 580001 67 123 190 265 348 438 535 639 748
Table 1
Alpha level Critical value
020 Gif(04081 n(3=2)thorn 1)010 Gif(04981 n(3=2)thorn 1)005 Gif(05797 n(3=2)thorn 1)001 Gif(07473 n(3=2)thorn 1)
A New One-Sample Test for Goodness-of-Fit 185
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So the statistic A2 has the value 4 The probability under the nullhypothesis that the A2 statistic assumes a value gtfrac14 4 is 00286 So thenull hypothesis would be rejected The cumulative distribution functionfor this two-sample test statistic (with mfrac14 4 and nfrac14 6) looks like this
2 POWER OF THE A STATISTIC
In an article titled lsquolsquoEDF Statistics for Goodness-of-fit and SomeComparisonsrsquorsquo Stephens 1974 presented a chart comparing the powersof several different statistical tests The null hypothesis is that we havea uniform random number on the interval (0 1) There were sevenalternate distributions defined as follows
F FethxTHORN frac14 1 eth1 zTHORNk 0 ltfrac14 z ltfrac14 1 k frac14 15 2
G FethxTHORN frac14 2ethk1THORNzk 0 ltfrac14 z ltfrac14 5
FethxTHORN frac14 1 2ethk1THORNeth1 zTHORNk 5 ltfrac14 z ltfrac14 1 k frac14 15 2 3
H FethxTHORN frac14 05 2ethk1THORNeth5 zTHORNk 0 ltfrac14 z ltfrac14 5
FethxTHORN frac14 05thorn 2ethk1THORNethz 5THORNk 5 ltfrac14 z ltfrac14 1 k frac14 15 2
I computed the power of the A statistic in these same seven situationsThe results are shown in section four of this paper The power of theA statistic compares very favorably with both the Kolmogorov-SmirnovD statistic and the Cramer-von Mises W2 statistic
3 TABLES OF THE A STATISTIC
Tables of the exact distribution of A for sample sizes (n) rangingfrom 2 through 7 is shown in Table 3
The tables of the A statistic (Table 4) were computed using simula-tions The number of replications (N) is shown followed by the samplesize (n) and four critical values with the corresponding alpha levels
Value 0 1 2 3 4Cumulative density 1714 5143 8381 9714 1000
186 Damico
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In each case the four alpha levels were selected to be near 020 010 005and 001
Here are some additional critical values for values of n from 8through 15 (Tables 5 and 6)
4 TABLES COMPARING POWER OF THE
A STATISTIC WITH OTHER
ONE-SAMPLE TESTS
As noted earlier the table of results comparing Kolmogorov-Smirnov D Cramer-von Mises W2 Kuiper V Watson U2 Anderson-Darling A2 Q (frac14Si ln zi) and Chi2 first appeared in an article byStephens (1974) in JASA I have simply added the column showing thepower of the A statistic as estimated by a number of simulations
Table 3
A nfrac14 2 nfrac14 3 nfrac14 4 nfrac14 5 nfrac14 6 nfrac14 7
0 0500 022222 009375 003840 001543 000611991 1000 066667 037500 019200 009259 004283932 092593 068750 044160 025720 013973773 100000 087500 067200 046296 029273524 096094 082560 064472 046256235 099219 091520 077761 061317996 100000 096352 086703 073131097 098656 092438 081845158 099616 095945 088053459 099936 097990 0923662710 100000 099096 0952885311 099640 0972077912 099880 0984233913 099970 0991597214 099996 0995813215 100000 0998074216 0999198617 0999708518 0999912619 0999980620 0999997621 10000000
A New One-Sample Test for Goodness-of-Fit 187
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Table 4
N nCritical
value (A) P(A) gtfrac14A N nCritical
value (A) P(A) gtfrac14A
10000 8 9 02267 1000 18 32 019611 01183 38 010113 00543 44 004917 00089 57 0009
10000 9 11 02074 1000 19 33 020413 01176 40 010216 00462 48 004920 00106 63 0010
10000 10 13 02008 10000 20 37 0197316 00972 45 0099319 00406 53 0047724 00090 66 00092
10000 11 15 02018 1000 21 40 019719 00860 49 009521 00522 56 005127 00111 75 0009
10000 12 17 02015 1000 22 43 020020 01162 51 010024 00507 59 004930 00099 77 0010
10000 13 19 02056 1000 23 47 019823 01038 56 010027 00484 66 004935 00097 83 0010
10000 14 22 01909 1200 24 48 0200827 00945 58 0098331 00497 67 0049240 00085 87 00092
10000 15 23 02125 10000 25 51 0199129 00972 61 0103134 00468 72 0051543 00089 93 00107
1000 16 26 0192 1600 26 53 0203131 0092 65 0101336 0046 77 0048148 0010 97 00094
1000 17 29 0193 1600 27 57 0196934 0102 69 0098840 0045 80 0047551 0008 104 00100
188 Damico
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Table 4 Continued
N nCritical
value (A) P(A) gtfrac14A N nCritical
value (A) P(A) gtfrac14A
1600 28 59 01981 1600 38 96 02019
73 01019 118 0101386 00500 135 00500110 00100 181 00100
1600 29 66 01975 1600 39 99 0201380 01013 120 0100092 00481 140 00494117 00100 170 00094
10000 30 67 02029 10000 40 103 0201181 01025 126 0100095 00490 148 00496126 00095 191 00099
1600 31 71 01969 10000 50 145 0194587 00981 177 00984101 00500 205 00496128 00100 265 00099
1600 32 74 02038 10000 60 190 0199991 01000 232 00986107 00488 270 00498135 00100 348 00093
1600 33 77 01963 10000 70 240 0201990 01006 292 01027108 00500 340 00488141 00094 438 00095
1600 34 82 01975 10000 80 293 01988101 00988 357 01015117 00488 415 00530149 00100 535 00112
10000 35 84 02036 10000 90 349 01982102 01019 426 01015120 00493 495 00530152 00103 639 00112
1600 36 91 02019 10000 100 409 01930113 00981 499 00955132 00488 580 00467166 00100 748 00089
1600 37 90 01988110 01000128 00488172 00100
A New One-Sample Test for Goodness-of-Fit 189
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Table
5
nfrac148
nfrac149
nfrac1410
nfrac1411
Critical
value(A
)P(A
)gtfrac14A
Critical
value(A
)P(A
)gtfrac14A
Critical
value(A
)P(A
)gtfrac14A
Critical
value(A
)P(A
)gtfrac14A
902267
11
02074
13
02008
15
02018
10
01657
12
01580
14
01606
16
01628
11
01183
13
01176
15
01260
17
01336
12
00810
14
00874
16
00972
18
01091
13
00543
15
00633
17
00729
19
00860
14
00347
16
00462
18
00542
20
00670
15
00227
17
00335
19
00406
21
00522
16
00151
18
00241
20
00298
22
00417
17
00089
19
00157
21
00221
23
00334
20
00106
22
00166
24
00257
21
00062
23
00126
25
00194
24
00090
26
00147
27
00111
28
00082
190 Damico
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Table
6
nfrac1412
nfrac1413
nfrac1414
nfrac1415
Critical
value(A
)P(A
)gtfrac14A
Critical
value(A
)P(A
)gtfrac14A
Critical
value(A
)P(A
)gtfrac14A
Critical
value(A
)P(A
)gtfrac14A
17
02015
19
02056
21
02168
23
02125
18
01677
20
01733
22
01909
24
01873
19
01407
21
01488
23
01657
25
01648
20
01162
22
01249
24
01431
26
01454
21
00929
23
01038
25
01267
27
01285
22
00755
24
00878
26
01100
28
01132
23
00615
25
00736
27
00945
29
00972
24
00507
26
00606
28
00806
30
00841
25
00416
27
00484
29
00691
31
00723
26
00315
28
00410
30
00581
32
00624
27
00240
29
00334
31
00497
33
00542
28
00175
30
00278
32
00435
34
00468
29
00134
31
00221
33
00360
35
00403
30
00099
32
00184
34
00293
36
00344
33
00145
35
00236
37
00296
34
00120
36
00200
38
00240
35
00097
37
00175
39
00206
38
00141
40
00163
39
00119
41
00141
40
00085
42
00118
43
00089
A New One-Sample Test for Goodness-of-Fit 191
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(Table 7) The alternative distribution functions are described in sectiontwo of this paper The sample sizes (n) were set equal to 10 20 and 40The alpha level was set at 010
REFERENCES
Bradley J V (1968) Distribution-Free Statistical Tests EnglewoodCliffs NJ Prentice-Hall
Gibbons J D (1976) Nonparametric Methods for Quantitative AnalysisNew York Holt Rinehart and Winston
Gibbons J D (1985) Nonparametric Statistical Inference New YorkMarcel Dekker
Table 7
Alternative D W2 V U2 A2 Q Chi2 A
(nfrac14 10 alphafrac14 010)Fkfrac1415 023 027 018 019 024 043 ndash 027Fkfrac1420 054 060 035 035 058 ndash ndash 058Gkfrac1415 009 007 022 023 006 ndash ndash 007Gkfrac1420 009 007 040 044 006 ndash ndash 010Gkfrac1430 021 021 081 086 018 ndash ndash 029Hkfrac1415 ndash ndash ndash ndash ndash ndash ndash 014Hkfrac1420 ndash ndash ndash ndash ndash ndash ndash 021
(nfrac14 20 alphafrac14 010)Fkfrac1415 038 046 025 028 046 068 ndash 046Fkfrac1420 078 087 061 060 087 097 059 088Gkfrac1415 013 011 032 034 010 011 ndash 010Gkfrac1420 025 025 071 077 028 025 ndash 029Gkfrac1430 063 079 099 099 084 ndash ndash 083Hkfrac1415 025 020 036 037 028 ndash ndash 017Hkfrac1420 047 044 071 077 054 ndash ndash 036
(nfrac14 40 alphafrac14 010)Fkfrac1415 060 070 043 043 ndash 089 040 073Fkfrac1420 098 099 091 089 ndash 100 089 099Gkfrac1415 019 022 057 061 ndash ndash 039 023Gkfrac1420 056 072 096 098 ndash ndash 085 075Gkfrac1430 ndash ndash ndash ndash ndash ndash ndash 100Hkfrac1415 036 032 058 063 ndash ndash ndash 027Hkfrac1420 071 080 096 098 ndash ndash ndash 075
192 Damico
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Hollander M Wolfe D A (1973) Nonparametric Statistical Methods2nd ed New York Wiley
Stephens M A (1974) EDF statistics for goodness-of-fit and somecomparisons (In theory and methods) J Amer Stat Assoc Theoryand Methods Section September 69(347)730ndash737
A New One-Sample Test for Goodness-of-Fit 193
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z-scores and the corresponding values from the normal distribution(1824) 0034 (1344) 0089 (1298) 0097 (046) 0678 (088)0811 (096) 0832 and (106) 0855 We proceed as if we were testingthe null hypothesis that these seven decimal numbers (0034 through0855) came from a uniform distribution on the interval (000 100)We begin by defining seven intervals and determining the number ofobservations in each
(0000 0143) 3(0143 0286) 0(0286 0429) 0(0429 0571) 0(0571 0714) 1(0714 0857) 3(0857 1000) 0
We now determine the number of moves required to produce the lsquolsquogroundstatersquorsquo (ie one ball in each box)
1stmove
2ndmove
3rdmove
4thmove
5thmove
6thmove
(0000 0143) 3 2 2 1 1 1 1(0143 0286) 0 1 0 1 1 1 1(0286 0429) 0 0 1 1 1 1 1(0429 0571) 0 0 0 0 0 1 1(0571 0714) 1 1 1 1 2 1 1(0714 0857) 3 3 3 3 2 2 1(0857 1000) 0 0 0 0 0 0 1
So the computed value of the A statistic is 6 The probability under thenull hypothesis that the A statistic assumes a value gtfrac14 6 is 03868 Thisalpha level would generally not be considered significant and so the nullhypothesis would not be rejected
The mathematical description of the statistic is simply a summationof a number of terms We begin with a few definitions As already men-tioned the Xi ifrac14 1 n denote n observations from a population PxLet Si ifrac14 1 n denote the order statistics Let F(si) be the value of thecumulative distribution function for the ith order statistic Finally letGif( ) represent the greatest integer function We define A as follows
A frac14Xn
ifrac141
jGifethnFethsiTHORN thorn 1THORN ij
A New One-Sample Test for Goodness-of-Fit 183
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So for our example
Gifeth7 0034thorn 1THORN frac14 1 j1 1j frac14 0
Gifeth7 0089thorn 1THORN frac14 1 j1 2j frac14 1
Gifeth7 0097thorn 1THORN frac14 1 j1 3j frac14 2
Gifeth7 0678thorn 1THORN frac14 5 j5 4j frac14 1
Gifeth7 0812thorn 1THORN frac14 6 j6 5j frac14 1
Gifeth7 0834thorn 1THORN frac14 6 j6 6j frac14 0
Gifeth7 0855thorn 1THORN frac14 6 j6 7j frac14 1
And the value of A is 0thorn 1thorn 2thorn 1thorn 1thorn 0thorn 1frac14 6 as before It should beobvious that the absolute values of the seven differences computed aboveactually correspond to the number of lsquolsquomovesrsquorsquo required to place each ofthe n order statistics into its corresponding lsquolsquoboxrsquorsquo Thus the third orderstatistic which lsquolsquooriginated in box 1rsquorsquo must be moved twice in order tolsquolsquoend up in box 3rsquorsquo
This statistic shares with the chi-square the concept of partitioningthe range of the probability distribution However it has the decidedadvantage of being useful in situations where we are able to obtain onlya small sample
In section three of this article I have published the exact distributionof the A statistic for values of nfrac14 2 3 4 5 6 and 7 For larger values of nI used computer simulations to generate the approximate distributions
After completing this work I was able to compute some formulaethat give excellent predictions of the critical values for large values of n(These formulae were computed by using stepwise regression techniques)
The resulting formulae (for alpha levelsfrac14 01 05 10 and 20) aregiven in the Table 1 The values computed from these formulae fornfrac14 20 30 40 50 60 70 80 90 and 100 are given in Table 2
Obviously these are good estimations of the values computedthrough the simulation and appearing in section three of this article
Another interesting fact is that we could easily extend the definitionof this statistic to test the two-sample problem in the following way
Let Xi ifrac14 1 m denote m observations from a population Pxand let F(x) be the cdf Let yj jfrac14 1 n denote n observations froma second population Py and let G(y) be the cdf We wish to test the nullhypothesis Ffrac14G against the broad alternative hypothesis F not frac14 GAssume that m is less than or equal to n Let tk kfrac14 1 (mthorn n)be the order statistics from the combined samples We construct thetwo-sample statistic A2 using the ranks (rirsquos) of the original xirsquos in the
184 Damico
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combined sample
A2 frac14Xm
ifrac141
jGifethethriTHORN ethm=ethmthorn nthorn 1THORNTHORN thorn 1THORN ij
Example We have a sample of size 4 from the x population and asample of size 6 from the y population
The x observations (and ranks) are
089eth1THORN 097eth2THORN 125eth3THORN 498eth5THORN
The y observations (and ranks) are
134eth4THORN 678eth6THORN 714eth7THORN 834eth8THORN 912eth9THORN 934eth10THORN
We compute
Gifetheth1THORN eth4=11THORN thorn 1THORN frac14 1 j1 1j frac14 0
Gifetheth2THORN eth4=11THORN thorn 1THORN frac14 1 j1 2j frac14 1
Gifetheth3THORN eth4=11THORN thorn 1THORN frac14 2 j2 3j frac14 1
Gifetheth5THORN eth4=11THORN thorn 1THORN frac14 2 j2 4j frac14 2
Table 2
Values of n
Alpha levels 20 30 40 50 60 70 80 90 100
020 37 68 104 145 190 240 293 349 409010 45 82 127 177 232 292 357 426 499005 52 96 147 205 270 340 415 495 580001 67 123 190 265 348 438 535 639 748
Table 1
Alpha level Critical value
020 Gif(04081 n(3=2)thorn 1)010 Gif(04981 n(3=2)thorn 1)005 Gif(05797 n(3=2)thorn 1)001 Gif(07473 n(3=2)thorn 1)
A New One-Sample Test for Goodness-of-Fit 185
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So the statistic A2 has the value 4 The probability under the nullhypothesis that the A2 statistic assumes a value gtfrac14 4 is 00286 So thenull hypothesis would be rejected The cumulative distribution functionfor this two-sample test statistic (with mfrac14 4 and nfrac14 6) looks like this
2 POWER OF THE A STATISTIC
In an article titled lsquolsquoEDF Statistics for Goodness-of-fit and SomeComparisonsrsquorsquo Stephens 1974 presented a chart comparing the powersof several different statistical tests The null hypothesis is that we havea uniform random number on the interval (0 1) There were sevenalternate distributions defined as follows
F FethxTHORN frac14 1 eth1 zTHORNk 0 ltfrac14 z ltfrac14 1 k frac14 15 2
G FethxTHORN frac14 2ethk1THORNzk 0 ltfrac14 z ltfrac14 5
FethxTHORN frac14 1 2ethk1THORNeth1 zTHORNk 5 ltfrac14 z ltfrac14 1 k frac14 15 2 3
H FethxTHORN frac14 05 2ethk1THORNeth5 zTHORNk 0 ltfrac14 z ltfrac14 5
FethxTHORN frac14 05thorn 2ethk1THORNethz 5THORNk 5 ltfrac14 z ltfrac14 1 k frac14 15 2
I computed the power of the A statistic in these same seven situationsThe results are shown in section four of this paper The power of theA statistic compares very favorably with both the Kolmogorov-SmirnovD statistic and the Cramer-von Mises W2 statistic
3 TABLES OF THE A STATISTIC
Tables of the exact distribution of A for sample sizes (n) rangingfrom 2 through 7 is shown in Table 3
The tables of the A statistic (Table 4) were computed using simula-tions The number of replications (N) is shown followed by the samplesize (n) and four critical values with the corresponding alpha levels
Value 0 1 2 3 4Cumulative density 1714 5143 8381 9714 1000
186 Damico
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In each case the four alpha levels were selected to be near 020 010 005and 001
Here are some additional critical values for values of n from 8through 15 (Tables 5 and 6)
4 TABLES COMPARING POWER OF THE
A STATISTIC WITH OTHER
ONE-SAMPLE TESTS
As noted earlier the table of results comparing Kolmogorov-Smirnov D Cramer-von Mises W2 Kuiper V Watson U2 Anderson-Darling A2 Q (frac14Si ln zi) and Chi2 first appeared in an article byStephens (1974) in JASA I have simply added the column showing thepower of the A statistic as estimated by a number of simulations
Table 3
A nfrac14 2 nfrac14 3 nfrac14 4 nfrac14 5 nfrac14 6 nfrac14 7
0 0500 022222 009375 003840 001543 000611991 1000 066667 037500 019200 009259 004283932 092593 068750 044160 025720 013973773 100000 087500 067200 046296 029273524 096094 082560 064472 046256235 099219 091520 077761 061317996 100000 096352 086703 073131097 098656 092438 081845158 099616 095945 088053459 099936 097990 0923662710 100000 099096 0952885311 099640 0972077912 099880 0984233913 099970 0991597214 099996 0995813215 100000 0998074216 0999198617 0999708518 0999912619 0999980620 0999997621 10000000
A New One-Sample Test for Goodness-of-Fit 187
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Table 4
N nCritical
value (A) P(A) gtfrac14A N nCritical
value (A) P(A) gtfrac14A
10000 8 9 02267 1000 18 32 019611 01183 38 010113 00543 44 004917 00089 57 0009
10000 9 11 02074 1000 19 33 020413 01176 40 010216 00462 48 004920 00106 63 0010
10000 10 13 02008 10000 20 37 0197316 00972 45 0099319 00406 53 0047724 00090 66 00092
10000 11 15 02018 1000 21 40 019719 00860 49 009521 00522 56 005127 00111 75 0009
10000 12 17 02015 1000 22 43 020020 01162 51 010024 00507 59 004930 00099 77 0010
10000 13 19 02056 1000 23 47 019823 01038 56 010027 00484 66 004935 00097 83 0010
10000 14 22 01909 1200 24 48 0200827 00945 58 0098331 00497 67 0049240 00085 87 00092
10000 15 23 02125 10000 25 51 0199129 00972 61 0103134 00468 72 0051543 00089 93 00107
1000 16 26 0192 1600 26 53 0203131 0092 65 0101336 0046 77 0048148 0010 97 00094
1000 17 29 0193 1600 27 57 0196934 0102 69 0098840 0045 80 0047551 0008 104 00100
188 Damico
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Table 4 Continued
N nCritical
value (A) P(A) gtfrac14A N nCritical
value (A) P(A) gtfrac14A
1600 28 59 01981 1600 38 96 02019
73 01019 118 0101386 00500 135 00500110 00100 181 00100
1600 29 66 01975 1600 39 99 0201380 01013 120 0100092 00481 140 00494117 00100 170 00094
10000 30 67 02029 10000 40 103 0201181 01025 126 0100095 00490 148 00496126 00095 191 00099
1600 31 71 01969 10000 50 145 0194587 00981 177 00984101 00500 205 00496128 00100 265 00099
1600 32 74 02038 10000 60 190 0199991 01000 232 00986107 00488 270 00498135 00100 348 00093
1600 33 77 01963 10000 70 240 0201990 01006 292 01027108 00500 340 00488141 00094 438 00095
1600 34 82 01975 10000 80 293 01988101 00988 357 01015117 00488 415 00530149 00100 535 00112
10000 35 84 02036 10000 90 349 01982102 01019 426 01015120 00493 495 00530152 00103 639 00112
1600 36 91 02019 10000 100 409 01930113 00981 499 00955132 00488 580 00467166 00100 748 00089
1600 37 90 01988110 01000128 00488172 00100
A New One-Sample Test for Goodness-of-Fit 189
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Table
5
nfrac148
nfrac149
nfrac1410
nfrac1411
Critical
value(A
)P(A
)gtfrac14A
Critical
value(A
)P(A
)gtfrac14A
Critical
value(A
)P(A
)gtfrac14A
Critical
value(A
)P(A
)gtfrac14A
902267
11
02074
13
02008
15
02018
10
01657
12
01580
14
01606
16
01628
11
01183
13
01176
15
01260
17
01336
12
00810
14
00874
16
00972
18
01091
13
00543
15
00633
17
00729
19
00860
14
00347
16
00462
18
00542
20
00670
15
00227
17
00335
19
00406
21
00522
16
00151
18
00241
20
00298
22
00417
17
00089
19
00157
21
00221
23
00334
20
00106
22
00166
24
00257
21
00062
23
00126
25
00194
24
00090
26
00147
27
00111
28
00082
190 Damico
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Table
6
nfrac1412
nfrac1413
nfrac1414
nfrac1415
Critical
value(A
)P(A
)gtfrac14A
Critical
value(A
)P(A
)gtfrac14A
Critical
value(A
)P(A
)gtfrac14A
Critical
value(A
)P(A
)gtfrac14A
17
02015
19
02056
21
02168
23
02125
18
01677
20
01733
22
01909
24
01873
19
01407
21
01488
23
01657
25
01648
20
01162
22
01249
24
01431
26
01454
21
00929
23
01038
25
01267
27
01285
22
00755
24
00878
26
01100
28
01132
23
00615
25
00736
27
00945
29
00972
24
00507
26
00606
28
00806
30
00841
25
00416
27
00484
29
00691
31
00723
26
00315
28
00410
30
00581
32
00624
27
00240
29
00334
31
00497
33
00542
28
00175
30
00278
32
00435
34
00468
29
00134
31
00221
33
00360
35
00403
30
00099
32
00184
34
00293
36
00344
33
00145
35
00236
37
00296
34
00120
36
00200
38
00240
35
00097
37
00175
39
00206
38
00141
40
00163
39
00119
41
00141
40
00085
42
00118
43
00089
A New One-Sample Test for Goodness-of-Fit 191
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(Table 7) The alternative distribution functions are described in sectiontwo of this paper The sample sizes (n) were set equal to 10 20 and 40The alpha level was set at 010
REFERENCES
Bradley J V (1968) Distribution-Free Statistical Tests EnglewoodCliffs NJ Prentice-Hall
Gibbons J D (1976) Nonparametric Methods for Quantitative AnalysisNew York Holt Rinehart and Winston
Gibbons J D (1985) Nonparametric Statistical Inference New YorkMarcel Dekker
Table 7
Alternative D W2 V U2 A2 Q Chi2 A
(nfrac14 10 alphafrac14 010)Fkfrac1415 023 027 018 019 024 043 ndash 027Fkfrac1420 054 060 035 035 058 ndash ndash 058Gkfrac1415 009 007 022 023 006 ndash ndash 007Gkfrac1420 009 007 040 044 006 ndash ndash 010Gkfrac1430 021 021 081 086 018 ndash ndash 029Hkfrac1415 ndash ndash ndash ndash ndash ndash ndash 014Hkfrac1420 ndash ndash ndash ndash ndash ndash ndash 021
(nfrac14 20 alphafrac14 010)Fkfrac1415 038 046 025 028 046 068 ndash 046Fkfrac1420 078 087 061 060 087 097 059 088Gkfrac1415 013 011 032 034 010 011 ndash 010Gkfrac1420 025 025 071 077 028 025 ndash 029Gkfrac1430 063 079 099 099 084 ndash ndash 083Hkfrac1415 025 020 036 037 028 ndash ndash 017Hkfrac1420 047 044 071 077 054 ndash ndash 036
(nfrac14 40 alphafrac14 010)Fkfrac1415 060 070 043 043 ndash 089 040 073Fkfrac1420 098 099 091 089 ndash 100 089 099Gkfrac1415 019 022 057 061 ndash ndash 039 023Gkfrac1420 056 072 096 098 ndash ndash 085 075Gkfrac1430 ndash ndash ndash ndash ndash ndash ndash 100Hkfrac1415 036 032 058 063 ndash ndash ndash 027Hkfrac1420 071 080 096 098 ndash ndash ndash 075
192 Damico
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Hollander M Wolfe D A (1973) Nonparametric Statistical Methods2nd ed New York Wiley
Stephens M A (1974) EDF statistics for goodness-of-fit and somecomparisons (In theory and methods) J Amer Stat Assoc Theoryand Methods Section September 69(347)730ndash737
A New One-Sample Test for Goodness-of-Fit 193
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So for our example
Gifeth7 0034thorn 1THORN frac14 1 j1 1j frac14 0
Gifeth7 0089thorn 1THORN frac14 1 j1 2j frac14 1
Gifeth7 0097thorn 1THORN frac14 1 j1 3j frac14 2
Gifeth7 0678thorn 1THORN frac14 5 j5 4j frac14 1
Gifeth7 0812thorn 1THORN frac14 6 j6 5j frac14 1
Gifeth7 0834thorn 1THORN frac14 6 j6 6j frac14 0
Gifeth7 0855thorn 1THORN frac14 6 j6 7j frac14 1
And the value of A is 0thorn 1thorn 2thorn 1thorn 1thorn 0thorn 1frac14 6 as before It should beobvious that the absolute values of the seven differences computed aboveactually correspond to the number of lsquolsquomovesrsquorsquo required to place each ofthe n order statistics into its corresponding lsquolsquoboxrsquorsquo Thus the third orderstatistic which lsquolsquooriginated in box 1rsquorsquo must be moved twice in order tolsquolsquoend up in box 3rsquorsquo
This statistic shares with the chi-square the concept of partitioningthe range of the probability distribution However it has the decidedadvantage of being useful in situations where we are able to obtain onlya small sample
In section three of this article I have published the exact distributionof the A statistic for values of nfrac14 2 3 4 5 6 and 7 For larger values of nI used computer simulations to generate the approximate distributions
After completing this work I was able to compute some formulaethat give excellent predictions of the critical values for large values of n(These formulae were computed by using stepwise regression techniques)
The resulting formulae (for alpha levelsfrac14 01 05 10 and 20) aregiven in the Table 1 The values computed from these formulae fornfrac14 20 30 40 50 60 70 80 90 and 100 are given in Table 2
Obviously these are good estimations of the values computedthrough the simulation and appearing in section three of this article
Another interesting fact is that we could easily extend the definitionof this statistic to test the two-sample problem in the following way
Let Xi ifrac14 1 m denote m observations from a population Pxand let F(x) be the cdf Let yj jfrac14 1 n denote n observations froma second population Py and let G(y) be the cdf We wish to test the nullhypothesis Ffrac14G against the broad alternative hypothesis F not frac14 GAssume that m is less than or equal to n Let tk kfrac14 1 (mthorn n)be the order statistics from the combined samples We construct thetwo-sample statistic A2 using the ranks (rirsquos) of the original xirsquos in the
184 Damico
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combined sample
A2 frac14Xm
ifrac141
jGifethethriTHORN ethm=ethmthorn nthorn 1THORNTHORN thorn 1THORN ij
Example We have a sample of size 4 from the x population and asample of size 6 from the y population
The x observations (and ranks) are
089eth1THORN 097eth2THORN 125eth3THORN 498eth5THORN
The y observations (and ranks) are
134eth4THORN 678eth6THORN 714eth7THORN 834eth8THORN 912eth9THORN 934eth10THORN
We compute
Gifetheth1THORN eth4=11THORN thorn 1THORN frac14 1 j1 1j frac14 0
Gifetheth2THORN eth4=11THORN thorn 1THORN frac14 1 j1 2j frac14 1
Gifetheth3THORN eth4=11THORN thorn 1THORN frac14 2 j2 3j frac14 1
Gifetheth5THORN eth4=11THORN thorn 1THORN frac14 2 j2 4j frac14 2
Table 2
Values of n
Alpha levels 20 30 40 50 60 70 80 90 100
020 37 68 104 145 190 240 293 349 409010 45 82 127 177 232 292 357 426 499005 52 96 147 205 270 340 415 495 580001 67 123 190 265 348 438 535 639 748
Table 1
Alpha level Critical value
020 Gif(04081 n(3=2)thorn 1)010 Gif(04981 n(3=2)thorn 1)005 Gif(05797 n(3=2)thorn 1)001 Gif(07473 n(3=2)thorn 1)
A New One-Sample Test for Goodness-of-Fit 185
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ORDER REPRINTS
So the statistic A2 has the value 4 The probability under the nullhypothesis that the A2 statistic assumes a value gtfrac14 4 is 00286 So thenull hypothesis would be rejected The cumulative distribution functionfor this two-sample test statistic (with mfrac14 4 and nfrac14 6) looks like this
2 POWER OF THE A STATISTIC
In an article titled lsquolsquoEDF Statistics for Goodness-of-fit and SomeComparisonsrsquorsquo Stephens 1974 presented a chart comparing the powersof several different statistical tests The null hypothesis is that we havea uniform random number on the interval (0 1) There were sevenalternate distributions defined as follows
F FethxTHORN frac14 1 eth1 zTHORNk 0 ltfrac14 z ltfrac14 1 k frac14 15 2
G FethxTHORN frac14 2ethk1THORNzk 0 ltfrac14 z ltfrac14 5
FethxTHORN frac14 1 2ethk1THORNeth1 zTHORNk 5 ltfrac14 z ltfrac14 1 k frac14 15 2 3
H FethxTHORN frac14 05 2ethk1THORNeth5 zTHORNk 0 ltfrac14 z ltfrac14 5
FethxTHORN frac14 05thorn 2ethk1THORNethz 5THORNk 5 ltfrac14 z ltfrac14 1 k frac14 15 2
I computed the power of the A statistic in these same seven situationsThe results are shown in section four of this paper The power of theA statistic compares very favorably with both the Kolmogorov-SmirnovD statistic and the Cramer-von Mises W2 statistic
3 TABLES OF THE A STATISTIC
Tables of the exact distribution of A for sample sizes (n) rangingfrom 2 through 7 is shown in Table 3
The tables of the A statistic (Table 4) were computed using simula-tions The number of replications (N) is shown followed by the samplesize (n) and four critical values with the corresponding alpha levels
Value 0 1 2 3 4Cumulative density 1714 5143 8381 9714 1000
186 Damico
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In each case the four alpha levels were selected to be near 020 010 005and 001
Here are some additional critical values for values of n from 8through 15 (Tables 5 and 6)
4 TABLES COMPARING POWER OF THE
A STATISTIC WITH OTHER
ONE-SAMPLE TESTS
As noted earlier the table of results comparing Kolmogorov-Smirnov D Cramer-von Mises W2 Kuiper V Watson U2 Anderson-Darling A2 Q (frac14Si ln zi) and Chi2 first appeared in an article byStephens (1974) in JASA I have simply added the column showing thepower of the A statistic as estimated by a number of simulations
Table 3
A nfrac14 2 nfrac14 3 nfrac14 4 nfrac14 5 nfrac14 6 nfrac14 7
0 0500 022222 009375 003840 001543 000611991 1000 066667 037500 019200 009259 004283932 092593 068750 044160 025720 013973773 100000 087500 067200 046296 029273524 096094 082560 064472 046256235 099219 091520 077761 061317996 100000 096352 086703 073131097 098656 092438 081845158 099616 095945 088053459 099936 097990 0923662710 100000 099096 0952885311 099640 0972077912 099880 0984233913 099970 0991597214 099996 0995813215 100000 0998074216 0999198617 0999708518 0999912619 0999980620 0999997621 10000000
A New One-Sample Test for Goodness-of-Fit 187
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Table 4
N nCritical
value (A) P(A) gtfrac14A N nCritical
value (A) P(A) gtfrac14A
10000 8 9 02267 1000 18 32 019611 01183 38 010113 00543 44 004917 00089 57 0009
10000 9 11 02074 1000 19 33 020413 01176 40 010216 00462 48 004920 00106 63 0010
10000 10 13 02008 10000 20 37 0197316 00972 45 0099319 00406 53 0047724 00090 66 00092
10000 11 15 02018 1000 21 40 019719 00860 49 009521 00522 56 005127 00111 75 0009
10000 12 17 02015 1000 22 43 020020 01162 51 010024 00507 59 004930 00099 77 0010
10000 13 19 02056 1000 23 47 019823 01038 56 010027 00484 66 004935 00097 83 0010
10000 14 22 01909 1200 24 48 0200827 00945 58 0098331 00497 67 0049240 00085 87 00092
10000 15 23 02125 10000 25 51 0199129 00972 61 0103134 00468 72 0051543 00089 93 00107
1000 16 26 0192 1600 26 53 0203131 0092 65 0101336 0046 77 0048148 0010 97 00094
1000 17 29 0193 1600 27 57 0196934 0102 69 0098840 0045 80 0047551 0008 104 00100
188 Damico
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Table 4 Continued
N nCritical
value (A) P(A) gtfrac14A N nCritical
value (A) P(A) gtfrac14A
1600 28 59 01981 1600 38 96 02019
73 01019 118 0101386 00500 135 00500110 00100 181 00100
1600 29 66 01975 1600 39 99 0201380 01013 120 0100092 00481 140 00494117 00100 170 00094
10000 30 67 02029 10000 40 103 0201181 01025 126 0100095 00490 148 00496126 00095 191 00099
1600 31 71 01969 10000 50 145 0194587 00981 177 00984101 00500 205 00496128 00100 265 00099
1600 32 74 02038 10000 60 190 0199991 01000 232 00986107 00488 270 00498135 00100 348 00093
1600 33 77 01963 10000 70 240 0201990 01006 292 01027108 00500 340 00488141 00094 438 00095
1600 34 82 01975 10000 80 293 01988101 00988 357 01015117 00488 415 00530149 00100 535 00112
10000 35 84 02036 10000 90 349 01982102 01019 426 01015120 00493 495 00530152 00103 639 00112
1600 36 91 02019 10000 100 409 01930113 00981 499 00955132 00488 580 00467166 00100 748 00089
1600 37 90 01988110 01000128 00488172 00100
A New One-Sample Test for Goodness-of-Fit 189
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Table
5
nfrac148
nfrac149
nfrac1410
nfrac1411
Critical
value(A
)P(A
)gtfrac14A
Critical
value(A
)P(A
)gtfrac14A
Critical
value(A
)P(A
)gtfrac14A
Critical
value(A
)P(A
)gtfrac14A
902267
11
02074
13
02008
15
02018
10
01657
12
01580
14
01606
16
01628
11
01183
13
01176
15
01260
17
01336
12
00810
14
00874
16
00972
18
01091
13
00543
15
00633
17
00729
19
00860
14
00347
16
00462
18
00542
20
00670
15
00227
17
00335
19
00406
21
00522
16
00151
18
00241
20
00298
22
00417
17
00089
19
00157
21
00221
23
00334
20
00106
22
00166
24
00257
21
00062
23
00126
25
00194
24
00090
26
00147
27
00111
28
00082
190 Damico
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Table
6
nfrac1412
nfrac1413
nfrac1414
nfrac1415
Critical
value(A
)P(A
)gtfrac14A
Critical
value(A
)P(A
)gtfrac14A
Critical
value(A
)P(A
)gtfrac14A
Critical
value(A
)P(A
)gtfrac14A
17
02015
19
02056
21
02168
23
02125
18
01677
20
01733
22
01909
24
01873
19
01407
21
01488
23
01657
25
01648
20
01162
22
01249
24
01431
26
01454
21
00929
23
01038
25
01267
27
01285
22
00755
24
00878
26
01100
28
01132
23
00615
25
00736
27
00945
29
00972
24
00507
26
00606
28
00806
30
00841
25
00416
27
00484
29
00691
31
00723
26
00315
28
00410
30
00581
32
00624
27
00240
29
00334
31
00497
33
00542
28
00175
30
00278
32
00435
34
00468
29
00134
31
00221
33
00360
35
00403
30
00099
32
00184
34
00293
36
00344
33
00145
35
00236
37
00296
34
00120
36
00200
38
00240
35
00097
37
00175
39
00206
38
00141
40
00163
39
00119
41
00141
40
00085
42
00118
43
00089
A New One-Sample Test for Goodness-of-Fit 191
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ORDER REPRINTS
(Table 7) The alternative distribution functions are described in sectiontwo of this paper The sample sizes (n) were set equal to 10 20 and 40The alpha level was set at 010
REFERENCES
Bradley J V (1968) Distribution-Free Statistical Tests EnglewoodCliffs NJ Prentice-Hall
Gibbons J D (1976) Nonparametric Methods for Quantitative AnalysisNew York Holt Rinehart and Winston
Gibbons J D (1985) Nonparametric Statistical Inference New YorkMarcel Dekker
Table 7
Alternative D W2 V U2 A2 Q Chi2 A
(nfrac14 10 alphafrac14 010)Fkfrac1415 023 027 018 019 024 043 ndash 027Fkfrac1420 054 060 035 035 058 ndash ndash 058Gkfrac1415 009 007 022 023 006 ndash ndash 007Gkfrac1420 009 007 040 044 006 ndash ndash 010Gkfrac1430 021 021 081 086 018 ndash ndash 029Hkfrac1415 ndash ndash ndash ndash ndash ndash ndash 014Hkfrac1420 ndash ndash ndash ndash ndash ndash ndash 021
(nfrac14 20 alphafrac14 010)Fkfrac1415 038 046 025 028 046 068 ndash 046Fkfrac1420 078 087 061 060 087 097 059 088Gkfrac1415 013 011 032 034 010 011 ndash 010Gkfrac1420 025 025 071 077 028 025 ndash 029Gkfrac1430 063 079 099 099 084 ndash ndash 083Hkfrac1415 025 020 036 037 028 ndash ndash 017Hkfrac1420 047 044 071 077 054 ndash ndash 036
(nfrac14 40 alphafrac14 010)Fkfrac1415 060 070 043 043 ndash 089 040 073Fkfrac1420 098 099 091 089 ndash 100 089 099Gkfrac1415 019 022 057 061 ndash ndash 039 023Gkfrac1420 056 072 096 098 ndash ndash 085 075Gkfrac1430 ndash ndash ndash ndash ndash ndash ndash 100Hkfrac1415 036 032 058 063 ndash ndash ndash 027Hkfrac1420 071 080 096 098 ndash ndash ndash 075
192 Damico
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a R
iver
side
Lib
rari
es]
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005
03
Nov
embe
r 20
14
ORDER REPRINTS
Hollander M Wolfe D A (1973) Nonparametric Statistical Methods2nd ed New York Wiley
Stephens M A (1974) EDF statistics for goodness-of-fit and somecomparisons (In theory and methods) J Amer Stat Assoc Theoryand Methods Section September 69(347)730ndash737
A New One-Sample Test for Goodness-of-Fit 193
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Nov
embe
r 20
14
Request PermissionOrder Reprints
Reprints of this article can also be ordered at
httpwwwdekkercomservletproductDOI101081STA120026585
Request Permission or Order Reprints Instantly
Interested in copying and sharing this article In most cases US Copyright Law requires that you get permission from the articlersquos rightsholder before using copyrighted content
All information and materials found in this article including but not limited to text trademarks patents logos graphics and images (the Materials) are the copyrighted works and other forms of intellectual property of Marcel Dekker Inc or its licensors All rights not expressly granted are reserved
Get permission to lawfully reproduce and distribute the Materials or order reprints quickly and painlessly Simply click on the Request Permission Order Reprints link below and follow the instructions Visit the US Copyright Office for information on Fair Use limitations of US copyright law Please refer to The Association of American Publishersrsquo (AAP) website for guidelines on Fair Use in the Classroom
The Materials are for your personal use only and cannot be reformatted reposted resold or distributed by electronic means or otherwise without permission from Marcel Dekker Inc Marcel Dekker Inc grants you the limited right to display the Materials only on your personal computer or personal wireless device and to copy and download single copies of such Materials provided that any copyright trademark or other notice appearing on such Materials is also retained by displayed copied or downloaded as part of the Materials and is not removed or obscured and provided you do not edit modify alter or enhance the Materials Please refer to our Website User Agreement for more details
Dow
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by [
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ORDER REPRINTS
combined sample
A2 frac14Xm
ifrac141
jGifethethriTHORN ethm=ethmthorn nthorn 1THORNTHORN thorn 1THORN ij
Example We have a sample of size 4 from the x population and asample of size 6 from the y population
The x observations (and ranks) are
089eth1THORN 097eth2THORN 125eth3THORN 498eth5THORN
The y observations (and ranks) are
134eth4THORN 678eth6THORN 714eth7THORN 834eth8THORN 912eth9THORN 934eth10THORN
We compute
Gifetheth1THORN eth4=11THORN thorn 1THORN frac14 1 j1 1j frac14 0
Gifetheth2THORN eth4=11THORN thorn 1THORN frac14 1 j1 2j frac14 1
Gifetheth3THORN eth4=11THORN thorn 1THORN frac14 2 j2 3j frac14 1
Gifetheth5THORN eth4=11THORN thorn 1THORN frac14 2 j2 4j frac14 2
Table 2
Values of n
Alpha levels 20 30 40 50 60 70 80 90 100
020 37 68 104 145 190 240 293 349 409010 45 82 127 177 232 292 357 426 499005 52 96 147 205 270 340 415 495 580001 67 123 190 265 348 438 535 639 748
Table 1
Alpha level Critical value
020 Gif(04081 n(3=2)thorn 1)010 Gif(04981 n(3=2)thorn 1)005 Gif(05797 n(3=2)thorn 1)001 Gif(07473 n(3=2)thorn 1)
A New One-Sample Test for Goodness-of-Fit 185
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So the statistic A2 has the value 4 The probability under the nullhypothesis that the A2 statistic assumes a value gtfrac14 4 is 00286 So thenull hypothesis would be rejected The cumulative distribution functionfor this two-sample test statistic (with mfrac14 4 and nfrac14 6) looks like this
2 POWER OF THE A STATISTIC
In an article titled lsquolsquoEDF Statistics for Goodness-of-fit and SomeComparisonsrsquorsquo Stephens 1974 presented a chart comparing the powersof several different statistical tests The null hypothesis is that we havea uniform random number on the interval (0 1) There were sevenalternate distributions defined as follows
F FethxTHORN frac14 1 eth1 zTHORNk 0 ltfrac14 z ltfrac14 1 k frac14 15 2
G FethxTHORN frac14 2ethk1THORNzk 0 ltfrac14 z ltfrac14 5
FethxTHORN frac14 1 2ethk1THORNeth1 zTHORNk 5 ltfrac14 z ltfrac14 1 k frac14 15 2 3
H FethxTHORN frac14 05 2ethk1THORNeth5 zTHORNk 0 ltfrac14 z ltfrac14 5
FethxTHORN frac14 05thorn 2ethk1THORNethz 5THORNk 5 ltfrac14 z ltfrac14 1 k frac14 15 2
I computed the power of the A statistic in these same seven situationsThe results are shown in section four of this paper The power of theA statistic compares very favorably with both the Kolmogorov-SmirnovD statistic and the Cramer-von Mises W2 statistic
3 TABLES OF THE A STATISTIC
Tables of the exact distribution of A for sample sizes (n) rangingfrom 2 through 7 is shown in Table 3
The tables of the A statistic (Table 4) were computed using simula-tions The number of replications (N) is shown followed by the samplesize (n) and four critical values with the corresponding alpha levels
Value 0 1 2 3 4Cumulative density 1714 5143 8381 9714 1000
186 Damico
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embe
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ORDER REPRINTS
In each case the four alpha levels were selected to be near 020 010 005and 001
Here are some additional critical values for values of n from 8through 15 (Tables 5 and 6)
4 TABLES COMPARING POWER OF THE
A STATISTIC WITH OTHER
ONE-SAMPLE TESTS
As noted earlier the table of results comparing Kolmogorov-Smirnov D Cramer-von Mises W2 Kuiper V Watson U2 Anderson-Darling A2 Q (frac14Si ln zi) and Chi2 first appeared in an article byStephens (1974) in JASA I have simply added the column showing thepower of the A statistic as estimated by a number of simulations
Table 3
A nfrac14 2 nfrac14 3 nfrac14 4 nfrac14 5 nfrac14 6 nfrac14 7
0 0500 022222 009375 003840 001543 000611991 1000 066667 037500 019200 009259 004283932 092593 068750 044160 025720 013973773 100000 087500 067200 046296 029273524 096094 082560 064472 046256235 099219 091520 077761 061317996 100000 096352 086703 073131097 098656 092438 081845158 099616 095945 088053459 099936 097990 0923662710 100000 099096 0952885311 099640 0972077912 099880 0984233913 099970 0991597214 099996 0995813215 100000 0998074216 0999198617 0999708518 0999912619 0999980620 0999997621 10000000
A New One-Sample Test for Goodness-of-Fit 187
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Table 4
N nCritical
value (A) P(A) gtfrac14A N nCritical
value (A) P(A) gtfrac14A
10000 8 9 02267 1000 18 32 019611 01183 38 010113 00543 44 004917 00089 57 0009
10000 9 11 02074 1000 19 33 020413 01176 40 010216 00462 48 004920 00106 63 0010
10000 10 13 02008 10000 20 37 0197316 00972 45 0099319 00406 53 0047724 00090 66 00092
10000 11 15 02018 1000 21 40 019719 00860 49 009521 00522 56 005127 00111 75 0009
10000 12 17 02015 1000 22 43 020020 01162 51 010024 00507 59 004930 00099 77 0010
10000 13 19 02056 1000 23 47 019823 01038 56 010027 00484 66 004935 00097 83 0010
10000 14 22 01909 1200 24 48 0200827 00945 58 0098331 00497 67 0049240 00085 87 00092
10000 15 23 02125 10000 25 51 0199129 00972 61 0103134 00468 72 0051543 00089 93 00107
1000 16 26 0192 1600 26 53 0203131 0092 65 0101336 0046 77 0048148 0010 97 00094
1000 17 29 0193 1600 27 57 0196934 0102 69 0098840 0045 80 0047551 0008 104 00100
188 Damico
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Table 4 Continued
N nCritical
value (A) P(A) gtfrac14A N nCritical
value (A) P(A) gtfrac14A
1600 28 59 01981 1600 38 96 02019
73 01019 118 0101386 00500 135 00500110 00100 181 00100
1600 29 66 01975 1600 39 99 0201380 01013 120 0100092 00481 140 00494117 00100 170 00094
10000 30 67 02029 10000 40 103 0201181 01025 126 0100095 00490 148 00496126 00095 191 00099
1600 31 71 01969 10000 50 145 0194587 00981 177 00984101 00500 205 00496128 00100 265 00099
1600 32 74 02038 10000 60 190 0199991 01000 232 00986107 00488 270 00498135 00100 348 00093
1600 33 77 01963 10000 70 240 0201990 01006 292 01027108 00500 340 00488141 00094 438 00095
1600 34 82 01975 10000 80 293 01988101 00988 357 01015117 00488 415 00530149 00100 535 00112
10000 35 84 02036 10000 90 349 01982102 01019 426 01015120 00493 495 00530152 00103 639 00112
1600 36 91 02019 10000 100 409 01930113 00981 499 00955132 00488 580 00467166 00100 748 00089
1600 37 90 01988110 01000128 00488172 00100
A New One-Sample Test for Goodness-of-Fit 189
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Table
5
nfrac148
nfrac149
nfrac1410
nfrac1411
Critical
value(A
)P(A
)gtfrac14A
Critical
value(A
)P(A
)gtfrac14A
Critical
value(A
)P(A
)gtfrac14A
Critical
value(A
)P(A
)gtfrac14A
902267
11
02074
13
02008
15
02018
10
01657
12
01580
14
01606
16
01628
11
01183
13
01176
15
01260
17
01336
12
00810
14
00874
16
00972
18
01091
13
00543
15
00633
17
00729
19
00860
14
00347
16
00462
18
00542
20
00670
15
00227
17
00335
19
00406
21
00522
16
00151
18
00241
20
00298
22
00417
17
00089
19
00157
21
00221
23
00334
20
00106
22
00166
24
00257
21
00062
23
00126
25
00194
24
00090
26
00147
27
00111
28
00082
190 Damico
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Table
6
nfrac1412
nfrac1413
nfrac1414
nfrac1415
Critical
value(A
)P(A
)gtfrac14A
Critical
value(A
)P(A
)gtfrac14A
Critical
value(A
)P(A
)gtfrac14A
Critical
value(A
)P(A
)gtfrac14A
17
02015
19
02056
21
02168
23
02125
18
01677
20
01733
22
01909
24
01873
19
01407
21
01488
23
01657
25
01648
20
01162
22
01249
24
01431
26
01454
21
00929
23
01038
25
01267
27
01285
22
00755
24
00878
26
01100
28
01132
23
00615
25
00736
27
00945
29
00972
24
00507
26
00606
28
00806
30
00841
25
00416
27
00484
29
00691
31
00723
26
00315
28
00410
30
00581
32
00624
27
00240
29
00334
31
00497
33
00542
28
00175
30
00278
32
00435
34
00468
29
00134
31
00221
33
00360
35
00403
30
00099
32
00184
34
00293
36
00344
33
00145
35
00236
37
00296
34
00120
36
00200
38
00240
35
00097
37
00175
39
00206
38
00141
40
00163
39
00119
41
00141
40
00085
42
00118
43
00089
A New One-Sample Test for Goodness-of-Fit 191
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Nov
embe
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ORDER REPRINTS
(Table 7) The alternative distribution functions are described in sectiontwo of this paper The sample sizes (n) were set equal to 10 20 and 40The alpha level was set at 010
REFERENCES
Bradley J V (1968) Distribution-Free Statistical Tests EnglewoodCliffs NJ Prentice-Hall
Gibbons J D (1976) Nonparametric Methods for Quantitative AnalysisNew York Holt Rinehart and Winston
Gibbons J D (1985) Nonparametric Statistical Inference New YorkMarcel Dekker
Table 7
Alternative D W2 V U2 A2 Q Chi2 A
(nfrac14 10 alphafrac14 010)Fkfrac1415 023 027 018 019 024 043 ndash 027Fkfrac1420 054 060 035 035 058 ndash ndash 058Gkfrac1415 009 007 022 023 006 ndash ndash 007Gkfrac1420 009 007 040 044 006 ndash ndash 010Gkfrac1430 021 021 081 086 018 ndash ndash 029Hkfrac1415 ndash ndash ndash ndash ndash ndash ndash 014Hkfrac1420 ndash ndash ndash ndash ndash ndash ndash 021
(nfrac14 20 alphafrac14 010)Fkfrac1415 038 046 025 028 046 068 ndash 046Fkfrac1420 078 087 061 060 087 097 059 088Gkfrac1415 013 011 032 034 010 011 ndash 010Gkfrac1420 025 025 071 077 028 025 ndash 029Gkfrac1430 063 079 099 099 084 ndash ndash 083Hkfrac1415 025 020 036 037 028 ndash ndash 017Hkfrac1420 047 044 071 077 054 ndash ndash 036
(nfrac14 40 alphafrac14 010)Fkfrac1415 060 070 043 043 ndash 089 040 073Fkfrac1420 098 099 091 089 ndash 100 089 099Gkfrac1415 019 022 057 061 ndash ndash 039 023Gkfrac1420 056 072 096 098 ndash ndash 085 075Gkfrac1430 ndash ndash ndash ndash ndash ndash ndash 100Hkfrac1415 036 032 058 063 ndash ndash ndash 027Hkfrac1420 071 080 096 098 ndash ndash ndash 075
192 Damico
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embe
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Hollander M Wolfe D A (1973) Nonparametric Statistical Methods2nd ed New York Wiley
Stephens M A (1974) EDF statistics for goodness-of-fit and somecomparisons (In theory and methods) J Amer Stat Assoc Theoryand Methods Section September 69(347)730ndash737
A New One-Sample Test for Goodness-of-Fit 193
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embe
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Request PermissionOrder Reprints
Reprints of this article can also be ordered at
httpwwwdekkercomservletproductDOI101081STA120026585
Request Permission or Order Reprints Instantly
Interested in copying and sharing this article In most cases US Copyright Law requires that you get permission from the articlersquos rightsholder before using copyrighted content
All information and materials found in this article including but not limited to text trademarks patents logos graphics and images (the Materials) are the copyrighted works and other forms of intellectual property of Marcel Dekker Inc or its licensors All rights not expressly granted are reserved
Get permission to lawfully reproduce and distribute the Materials or order reprints quickly and painlessly Simply click on the Request Permission Order Reprints link below and follow the instructions Visit the US Copyright Office for information on Fair Use limitations of US copyright law Please refer to The Association of American Publishersrsquo (AAP) website for guidelines on Fair Use in the Classroom
The Materials are for your personal use only and cannot be reformatted reposted resold or distributed by electronic means or otherwise without permission from Marcel Dekker Inc Marcel Dekker Inc grants you the limited right to display the Materials only on your personal computer or personal wireless device and to copy and download single copies of such Materials provided that any copyright trademark or other notice appearing on such Materials is also retained by displayed copied or downloaded as part of the Materials and is not removed or obscured and provided you do not edit modify alter or enhance the Materials Please refer to our Website User Agreement for more details
Dow
nloa
ded
by [
Uni
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embe
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ORDER REPRINTS
So the statistic A2 has the value 4 The probability under the nullhypothesis that the A2 statistic assumes a value gtfrac14 4 is 00286 So thenull hypothesis would be rejected The cumulative distribution functionfor this two-sample test statistic (with mfrac14 4 and nfrac14 6) looks like this
2 POWER OF THE A STATISTIC
In an article titled lsquolsquoEDF Statistics for Goodness-of-fit and SomeComparisonsrsquorsquo Stephens 1974 presented a chart comparing the powersof several different statistical tests The null hypothesis is that we havea uniform random number on the interval (0 1) There were sevenalternate distributions defined as follows
F FethxTHORN frac14 1 eth1 zTHORNk 0 ltfrac14 z ltfrac14 1 k frac14 15 2
G FethxTHORN frac14 2ethk1THORNzk 0 ltfrac14 z ltfrac14 5
FethxTHORN frac14 1 2ethk1THORNeth1 zTHORNk 5 ltfrac14 z ltfrac14 1 k frac14 15 2 3
H FethxTHORN frac14 05 2ethk1THORNeth5 zTHORNk 0 ltfrac14 z ltfrac14 5
FethxTHORN frac14 05thorn 2ethk1THORNethz 5THORNk 5 ltfrac14 z ltfrac14 1 k frac14 15 2
I computed the power of the A statistic in these same seven situationsThe results are shown in section four of this paper The power of theA statistic compares very favorably with both the Kolmogorov-SmirnovD statistic and the Cramer-von Mises W2 statistic
3 TABLES OF THE A STATISTIC
Tables of the exact distribution of A for sample sizes (n) rangingfrom 2 through 7 is shown in Table 3
The tables of the A statistic (Table 4) were computed using simula-tions The number of replications (N) is shown followed by the samplesize (n) and four critical values with the corresponding alpha levels
Value 0 1 2 3 4Cumulative density 1714 5143 8381 9714 1000
186 Damico
Dow
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embe
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ORDER REPRINTS
In each case the four alpha levels were selected to be near 020 010 005and 001
Here are some additional critical values for values of n from 8through 15 (Tables 5 and 6)
4 TABLES COMPARING POWER OF THE
A STATISTIC WITH OTHER
ONE-SAMPLE TESTS
As noted earlier the table of results comparing Kolmogorov-Smirnov D Cramer-von Mises W2 Kuiper V Watson U2 Anderson-Darling A2 Q (frac14Si ln zi) and Chi2 first appeared in an article byStephens (1974) in JASA I have simply added the column showing thepower of the A statistic as estimated by a number of simulations
Table 3
A nfrac14 2 nfrac14 3 nfrac14 4 nfrac14 5 nfrac14 6 nfrac14 7
0 0500 022222 009375 003840 001543 000611991 1000 066667 037500 019200 009259 004283932 092593 068750 044160 025720 013973773 100000 087500 067200 046296 029273524 096094 082560 064472 046256235 099219 091520 077761 061317996 100000 096352 086703 073131097 098656 092438 081845158 099616 095945 088053459 099936 097990 0923662710 100000 099096 0952885311 099640 0972077912 099880 0984233913 099970 0991597214 099996 0995813215 100000 0998074216 0999198617 0999708518 0999912619 0999980620 0999997621 10000000
A New One-Sample Test for Goodness-of-Fit 187
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ORDER REPRINTS
Table 4
N nCritical
value (A) P(A) gtfrac14A N nCritical
value (A) P(A) gtfrac14A
10000 8 9 02267 1000 18 32 019611 01183 38 010113 00543 44 004917 00089 57 0009
10000 9 11 02074 1000 19 33 020413 01176 40 010216 00462 48 004920 00106 63 0010
10000 10 13 02008 10000 20 37 0197316 00972 45 0099319 00406 53 0047724 00090 66 00092
10000 11 15 02018 1000 21 40 019719 00860 49 009521 00522 56 005127 00111 75 0009
10000 12 17 02015 1000 22 43 020020 01162 51 010024 00507 59 004930 00099 77 0010
10000 13 19 02056 1000 23 47 019823 01038 56 010027 00484 66 004935 00097 83 0010
10000 14 22 01909 1200 24 48 0200827 00945 58 0098331 00497 67 0049240 00085 87 00092
10000 15 23 02125 10000 25 51 0199129 00972 61 0103134 00468 72 0051543 00089 93 00107
1000 16 26 0192 1600 26 53 0203131 0092 65 0101336 0046 77 0048148 0010 97 00094
1000 17 29 0193 1600 27 57 0196934 0102 69 0098840 0045 80 0047551 0008 104 00100
188 Damico
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ORDER REPRINTS
Table 4 Continued
N nCritical
value (A) P(A) gtfrac14A N nCritical
value (A) P(A) gtfrac14A
1600 28 59 01981 1600 38 96 02019
73 01019 118 0101386 00500 135 00500110 00100 181 00100
1600 29 66 01975 1600 39 99 0201380 01013 120 0100092 00481 140 00494117 00100 170 00094
10000 30 67 02029 10000 40 103 0201181 01025 126 0100095 00490 148 00496126 00095 191 00099
1600 31 71 01969 10000 50 145 0194587 00981 177 00984101 00500 205 00496128 00100 265 00099
1600 32 74 02038 10000 60 190 0199991 01000 232 00986107 00488 270 00498135 00100 348 00093
1600 33 77 01963 10000 70 240 0201990 01006 292 01027108 00500 340 00488141 00094 438 00095
1600 34 82 01975 10000 80 293 01988101 00988 357 01015117 00488 415 00530149 00100 535 00112
10000 35 84 02036 10000 90 349 01982102 01019 426 01015120 00493 495 00530152 00103 639 00112
1600 36 91 02019 10000 100 409 01930113 00981 499 00955132 00488 580 00467166 00100 748 00089
1600 37 90 01988110 01000128 00488172 00100
A New One-Sample Test for Goodness-of-Fit 189
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Table
5
nfrac148
nfrac149
nfrac1410
nfrac1411
Critical
value(A
)P(A
)gtfrac14A
Critical
value(A
)P(A
)gtfrac14A
Critical
value(A
)P(A
)gtfrac14A
Critical
value(A
)P(A
)gtfrac14A
902267
11
02074
13
02008
15
02018
10
01657
12
01580
14
01606
16
01628
11
01183
13
01176
15
01260
17
01336
12
00810
14
00874
16
00972
18
01091
13
00543
15
00633
17
00729
19
00860
14
00347
16
00462
18
00542
20
00670
15
00227
17
00335
19
00406
21
00522
16
00151
18
00241
20
00298
22
00417
17
00089
19
00157
21
00221
23
00334
20
00106
22
00166
24
00257
21
00062
23
00126
25
00194
24
00090
26
00147
27
00111
28
00082
190 Damico
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ORDER REPRINTS
Table
6
nfrac1412
nfrac1413
nfrac1414
nfrac1415
Critical
value(A
)P(A
)gtfrac14A
Critical
value(A
)P(A
)gtfrac14A
Critical
value(A
)P(A
)gtfrac14A
Critical
value(A
)P(A
)gtfrac14A
17
02015
19
02056
21
02168
23
02125
18
01677
20
01733
22
01909
24
01873
19
01407
21
01488
23
01657
25
01648
20
01162
22
01249
24
01431
26
01454
21
00929
23
01038
25
01267
27
01285
22
00755
24
00878
26
01100
28
01132
23
00615
25
00736
27
00945
29
00972
24
00507
26
00606
28
00806
30
00841
25
00416
27
00484
29
00691
31
00723
26
00315
28
00410
30
00581
32
00624
27
00240
29
00334
31
00497
33
00542
28
00175
30
00278
32
00435
34
00468
29
00134
31
00221
33
00360
35
00403
30
00099
32
00184
34
00293
36
00344
33
00145
35
00236
37
00296
34
00120
36
00200
38
00240
35
00097
37
00175
39
00206
38
00141
40
00163
39
00119
41
00141
40
00085
42
00118
43
00089
A New One-Sample Test for Goodness-of-Fit 191
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ORDER REPRINTS
(Table 7) The alternative distribution functions are described in sectiontwo of this paper The sample sizes (n) were set equal to 10 20 and 40The alpha level was set at 010
REFERENCES
Bradley J V (1968) Distribution-Free Statistical Tests EnglewoodCliffs NJ Prentice-Hall
Gibbons J D (1976) Nonparametric Methods for Quantitative AnalysisNew York Holt Rinehart and Winston
Gibbons J D (1985) Nonparametric Statistical Inference New YorkMarcel Dekker
Table 7
Alternative D W2 V U2 A2 Q Chi2 A
(nfrac14 10 alphafrac14 010)Fkfrac1415 023 027 018 019 024 043 ndash 027Fkfrac1420 054 060 035 035 058 ndash ndash 058Gkfrac1415 009 007 022 023 006 ndash ndash 007Gkfrac1420 009 007 040 044 006 ndash ndash 010Gkfrac1430 021 021 081 086 018 ndash ndash 029Hkfrac1415 ndash ndash ndash ndash ndash ndash ndash 014Hkfrac1420 ndash ndash ndash ndash ndash ndash ndash 021
(nfrac14 20 alphafrac14 010)Fkfrac1415 038 046 025 028 046 068 ndash 046Fkfrac1420 078 087 061 060 087 097 059 088Gkfrac1415 013 011 032 034 010 011 ndash 010Gkfrac1420 025 025 071 077 028 025 ndash 029Gkfrac1430 063 079 099 099 084 ndash ndash 083Hkfrac1415 025 020 036 037 028 ndash ndash 017Hkfrac1420 047 044 071 077 054 ndash ndash 036
(nfrac14 40 alphafrac14 010)Fkfrac1415 060 070 043 043 ndash 089 040 073Fkfrac1420 098 099 091 089 ndash 100 089 099Gkfrac1415 019 022 057 061 ndash ndash 039 023Gkfrac1420 056 072 096 098 ndash ndash 085 075Gkfrac1430 ndash ndash ndash ndash ndash ndash ndash 100Hkfrac1415 036 032 058 063 ndash ndash ndash 027Hkfrac1420 071 080 096 098 ndash ndash ndash 075
192 Damico
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embe
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14
ORDER REPRINTS
Hollander M Wolfe D A (1973) Nonparametric Statistical Methods2nd ed New York Wiley
Stephens M A (1974) EDF statistics for goodness-of-fit and somecomparisons (In theory and methods) J Amer Stat Assoc Theoryand Methods Section September 69(347)730ndash737
A New One-Sample Test for Goodness-of-Fit 193
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embe
r 20
14
Request PermissionOrder Reprints
Reprints of this article can also be ordered at
httpwwwdekkercomservletproductDOI101081STA120026585
Request Permission or Order Reprints Instantly
Interested in copying and sharing this article In most cases US Copyright Law requires that you get permission from the articlersquos rightsholder before using copyrighted content
All information and materials found in this article including but not limited to text trademarks patents logos graphics and images (the Materials) are the copyrighted works and other forms of intellectual property of Marcel Dekker Inc or its licensors All rights not expressly granted are reserved
Get permission to lawfully reproduce and distribute the Materials or order reprints quickly and painlessly Simply click on the Request Permission Order Reprints link below and follow the instructions Visit the US Copyright Office for information on Fair Use limitations of US copyright law Please refer to The Association of American Publishersrsquo (AAP) website for guidelines on Fair Use in the Classroom
The Materials are for your personal use only and cannot be reformatted reposted resold or distributed by electronic means or otherwise without permission from Marcel Dekker Inc Marcel Dekker Inc grants you the limited right to display the Materials only on your personal computer or personal wireless device and to copy and download single copies of such Materials provided that any copyright trademark or other notice appearing on such Materials is also retained by displayed copied or downloaded as part of the Materials and is not removed or obscured and provided you do not edit modify alter or enhance the Materials Please refer to our Website User Agreement for more details
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In each case the four alpha levels were selected to be near 020 010 005and 001
Here are some additional critical values for values of n from 8through 15 (Tables 5 and 6)
4 TABLES COMPARING POWER OF THE
A STATISTIC WITH OTHER
ONE-SAMPLE TESTS
As noted earlier the table of results comparing Kolmogorov-Smirnov D Cramer-von Mises W2 Kuiper V Watson U2 Anderson-Darling A2 Q (frac14Si ln zi) and Chi2 first appeared in an article byStephens (1974) in JASA I have simply added the column showing thepower of the A statistic as estimated by a number of simulations
Table 3
A nfrac14 2 nfrac14 3 nfrac14 4 nfrac14 5 nfrac14 6 nfrac14 7
0 0500 022222 009375 003840 001543 000611991 1000 066667 037500 019200 009259 004283932 092593 068750 044160 025720 013973773 100000 087500 067200 046296 029273524 096094 082560 064472 046256235 099219 091520 077761 061317996 100000 096352 086703 073131097 098656 092438 081845158 099616 095945 088053459 099936 097990 0923662710 100000 099096 0952885311 099640 0972077912 099880 0984233913 099970 0991597214 099996 0995813215 100000 0998074216 0999198617 0999708518 0999912619 0999980620 0999997621 10000000
A New One-Sample Test for Goodness-of-Fit 187
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Table 4
N nCritical
value (A) P(A) gtfrac14A N nCritical
value (A) P(A) gtfrac14A
10000 8 9 02267 1000 18 32 019611 01183 38 010113 00543 44 004917 00089 57 0009
10000 9 11 02074 1000 19 33 020413 01176 40 010216 00462 48 004920 00106 63 0010
10000 10 13 02008 10000 20 37 0197316 00972 45 0099319 00406 53 0047724 00090 66 00092
10000 11 15 02018 1000 21 40 019719 00860 49 009521 00522 56 005127 00111 75 0009
10000 12 17 02015 1000 22 43 020020 01162 51 010024 00507 59 004930 00099 77 0010
10000 13 19 02056 1000 23 47 019823 01038 56 010027 00484 66 004935 00097 83 0010
10000 14 22 01909 1200 24 48 0200827 00945 58 0098331 00497 67 0049240 00085 87 00092
10000 15 23 02125 10000 25 51 0199129 00972 61 0103134 00468 72 0051543 00089 93 00107
1000 16 26 0192 1600 26 53 0203131 0092 65 0101336 0046 77 0048148 0010 97 00094
1000 17 29 0193 1600 27 57 0196934 0102 69 0098840 0045 80 0047551 0008 104 00100
188 Damico
Dow
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by [
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14
ORDER REPRINTS
Table 4 Continued
N nCritical
value (A) P(A) gtfrac14A N nCritical
value (A) P(A) gtfrac14A
1600 28 59 01981 1600 38 96 02019
73 01019 118 0101386 00500 135 00500110 00100 181 00100
1600 29 66 01975 1600 39 99 0201380 01013 120 0100092 00481 140 00494117 00100 170 00094
10000 30 67 02029 10000 40 103 0201181 01025 126 0100095 00490 148 00496126 00095 191 00099
1600 31 71 01969 10000 50 145 0194587 00981 177 00984101 00500 205 00496128 00100 265 00099
1600 32 74 02038 10000 60 190 0199991 01000 232 00986107 00488 270 00498135 00100 348 00093
1600 33 77 01963 10000 70 240 0201990 01006 292 01027108 00500 340 00488141 00094 438 00095
1600 34 82 01975 10000 80 293 01988101 00988 357 01015117 00488 415 00530149 00100 535 00112
10000 35 84 02036 10000 90 349 01982102 01019 426 01015120 00493 495 00530152 00103 639 00112
1600 36 91 02019 10000 100 409 01930113 00981 499 00955132 00488 580 00467166 00100 748 00089
1600 37 90 01988110 01000128 00488172 00100
A New One-Sample Test for Goodness-of-Fit 189
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embe
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ORDER REPRINTS
Table
5
nfrac148
nfrac149
nfrac1410
nfrac1411
Critical
value(A
)P(A
)gtfrac14A
Critical
value(A
)P(A
)gtfrac14A
Critical
value(A
)P(A
)gtfrac14A
Critical
value(A
)P(A
)gtfrac14A
902267
11
02074
13
02008
15
02018
10
01657
12
01580
14
01606
16
01628
11
01183
13
01176
15
01260
17
01336
12
00810
14
00874
16
00972
18
01091
13
00543
15
00633
17
00729
19
00860
14
00347
16
00462
18
00542
20
00670
15
00227
17
00335
19
00406
21
00522
16
00151
18
00241
20
00298
22
00417
17
00089
19
00157
21
00221
23
00334
20
00106
22
00166
24
00257
21
00062
23
00126
25
00194
24
00090
26
00147
27
00111
28
00082
190 Damico
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ORDER REPRINTS
Table
6
nfrac1412
nfrac1413
nfrac1414
nfrac1415
Critical
value(A
)P(A
)gtfrac14A
Critical
value(A
)P(A
)gtfrac14A
Critical
value(A
)P(A
)gtfrac14A
Critical
value(A
)P(A
)gtfrac14A
17
02015
19
02056
21
02168
23
02125
18
01677
20
01733
22
01909
24
01873
19
01407
21
01488
23
01657
25
01648
20
01162
22
01249
24
01431
26
01454
21
00929
23
01038
25
01267
27
01285
22
00755
24
00878
26
01100
28
01132
23
00615
25
00736
27
00945
29
00972
24
00507
26
00606
28
00806
30
00841
25
00416
27
00484
29
00691
31
00723
26
00315
28
00410
30
00581
32
00624
27
00240
29
00334
31
00497
33
00542
28
00175
30
00278
32
00435
34
00468
29
00134
31
00221
33
00360
35
00403
30
00099
32
00184
34
00293
36
00344
33
00145
35
00236
37
00296
34
00120
36
00200
38
00240
35
00097
37
00175
39
00206
38
00141
40
00163
39
00119
41
00141
40
00085
42
00118
43
00089
A New One-Sample Test for Goodness-of-Fit 191
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embe
r 20
14
ORDER REPRINTS
(Table 7) The alternative distribution functions are described in sectiontwo of this paper The sample sizes (n) were set equal to 10 20 and 40The alpha level was set at 010
REFERENCES
Bradley J V (1968) Distribution-Free Statistical Tests EnglewoodCliffs NJ Prentice-Hall
Gibbons J D (1976) Nonparametric Methods for Quantitative AnalysisNew York Holt Rinehart and Winston
Gibbons J D (1985) Nonparametric Statistical Inference New YorkMarcel Dekker
Table 7
Alternative D W2 V U2 A2 Q Chi2 A
(nfrac14 10 alphafrac14 010)Fkfrac1415 023 027 018 019 024 043 ndash 027Fkfrac1420 054 060 035 035 058 ndash ndash 058Gkfrac1415 009 007 022 023 006 ndash ndash 007Gkfrac1420 009 007 040 044 006 ndash ndash 010Gkfrac1430 021 021 081 086 018 ndash ndash 029Hkfrac1415 ndash ndash ndash ndash ndash ndash ndash 014Hkfrac1420 ndash ndash ndash ndash ndash ndash ndash 021
(nfrac14 20 alphafrac14 010)Fkfrac1415 038 046 025 028 046 068 ndash 046Fkfrac1420 078 087 061 060 087 097 059 088Gkfrac1415 013 011 032 034 010 011 ndash 010Gkfrac1420 025 025 071 077 028 025 ndash 029Gkfrac1430 063 079 099 099 084 ndash ndash 083Hkfrac1415 025 020 036 037 028 ndash ndash 017Hkfrac1420 047 044 071 077 054 ndash ndash 036
(nfrac14 40 alphafrac14 010)Fkfrac1415 060 070 043 043 ndash 089 040 073Fkfrac1420 098 099 091 089 ndash 100 089 099Gkfrac1415 019 022 057 061 ndash ndash 039 023Gkfrac1420 056 072 096 098 ndash ndash 085 075Gkfrac1430 ndash ndash ndash ndash ndash ndash ndash 100Hkfrac1415 036 032 058 063 ndash ndash ndash 027Hkfrac1420 071 080 096 098 ndash ndash ndash 075
192 Damico
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embe
r 20
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ORDER REPRINTS
Hollander M Wolfe D A (1973) Nonparametric Statistical Methods2nd ed New York Wiley
Stephens M A (1974) EDF statistics for goodness-of-fit and somecomparisons (In theory and methods) J Amer Stat Assoc Theoryand Methods Section September 69(347)730ndash737
A New One-Sample Test for Goodness-of-Fit 193
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at 1
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Nov
embe
r 20
14
Request PermissionOrder Reprints
Reprints of this article can also be ordered at
httpwwwdekkercomservletproductDOI101081STA120026585
Request Permission or Order Reprints Instantly
Interested in copying and sharing this article In most cases US Copyright Law requires that you get permission from the articlersquos rightsholder before using copyrighted content
All information and materials found in this article including but not limited to text trademarks patents logos graphics and images (the Materials) are the copyrighted works and other forms of intellectual property of Marcel Dekker Inc or its licensors All rights not expressly granted are reserved
Get permission to lawfully reproduce and distribute the Materials or order reprints quickly and painlessly Simply click on the Request Permission Order Reprints link below and follow the instructions Visit the US Copyright Office for information on Fair Use limitations of US copyright law Please refer to The Association of American Publishersrsquo (AAP) website for guidelines on Fair Use in the Classroom
The Materials are for your personal use only and cannot be reformatted reposted resold or distributed by electronic means or otherwise without permission from Marcel Dekker Inc Marcel Dekker Inc grants you the limited right to display the Materials only on your personal computer or personal wireless device and to copy and download single copies of such Materials provided that any copyright trademark or other notice appearing on such Materials is also retained by displayed copied or downloaded as part of the Materials and is not removed or obscured and provided you do not edit modify alter or enhance the Materials Please refer to our Website User Agreement for more details
Dow
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ORDER REPRINTS
Table 4
N nCritical
value (A) P(A) gtfrac14A N nCritical
value (A) P(A) gtfrac14A
10000 8 9 02267 1000 18 32 019611 01183 38 010113 00543 44 004917 00089 57 0009
10000 9 11 02074 1000 19 33 020413 01176 40 010216 00462 48 004920 00106 63 0010
10000 10 13 02008 10000 20 37 0197316 00972 45 0099319 00406 53 0047724 00090 66 00092
10000 11 15 02018 1000 21 40 019719 00860 49 009521 00522 56 005127 00111 75 0009
10000 12 17 02015 1000 22 43 020020 01162 51 010024 00507 59 004930 00099 77 0010
10000 13 19 02056 1000 23 47 019823 01038 56 010027 00484 66 004935 00097 83 0010
10000 14 22 01909 1200 24 48 0200827 00945 58 0098331 00497 67 0049240 00085 87 00092
10000 15 23 02125 10000 25 51 0199129 00972 61 0103134 00468 72 0051543 00089 93 00107
1000 16 26 0192 1600 26 53 0203131 0092 65 0101336 0046 77 0048148 0010 97 00094
1000 17 29 0193 1600 27 57 0196934 0102 69 0098840 0045 80 0047551 0008 104 00100
188 Damico
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Table 4 Continued
N nCritical
value (A) P(A) gtfrac14A N nCritical
value (A) P(A) gtfrac14A
1600 28 59 01981 1600 38 96 02019
73 01019 118 0101386 00500 135 00500110 00100 181 00100
1600 29 66 01975 1600 39 99 0201380 01013 120 0100092 00481 140 00494117 00100 170 00094
10000 30 67 02029 10000 40 103 0201181 01025 126 0100095 00490 148 00496126 00095 191 00099
1600 31 71 01969 10000 50 145 0194587 00981 177 00984101 00500 205 00496128 00100 265 00099
1600 32 74 02038 10000 60 190 0199991 01000 232 00986107 00488 270 00498135 00100 348 00093
1600 33 77 01963 10000 70 240 0201990 01006 292 01027108 00500 340 00488141 00094 438 00095
1600 34 82 01975 10000 80 293 01988101 00988 357 01015117 00488 415 00530149 00100 535 00112
10000 35 84 02036 10000 90 349 01982102 01019 426 01015120 00493 495 00530152 00103 639 00112
1600 36 91 02019 10000 100 409 01930113 00981 499 00955132 00488 580 00467166 00100 748 00089
1600 37 90 01988110 01000128 00488172 00100
A New One-Sample Test for Goodness-of-Fit 189
Dow
nloa
ded
by [
Uni
vers
ity o
f C
alif
orni
a R
iver
side
Lib
rari
es]
at 1
005
03
Nov
embe
r 20
14
ORDER REPRINTS
Table
5
nfrac148
nfrac149
nfrac1410
nfrac1411
Critical
value(A
)P(A
)gtfrac14A
Critical
value(A
)P(A
)gtfrac14A
Critical
value(A
)P(A
)gtfrac14A
Critical
value(A
)P(A
)gtfrac14A
902267
11
02074
13
02008
15
02018
10
01657
12
01580
14
01606
16
01628
11
01183
13
01176
15
01260
17
01336
12
00810
14
00874
16
00972
18
01091
13
00543
15
00633
17
00729
19
00860
14
00347
16
00462
18
00542
20
00670
15
00227
17
00335
19
00406
21
00522
16
00151
18
00241
20
00298
22
00417
17
00089
19
00157
21
00221
23
00334
20
00106
22
00166
24
00257
21
00062
23
00126
25
00194
24
00090
26
00147
27
00111
28
00082
190 Damico
Dow
nloa
ded
by [
Uni
vers
ity o
f C
alif
orni
a R
iver
side
Lib
rari
es]
at 1
005
03
Nov
embe
r 20
14
ORDER REPRINTS
Table
6
nfrac1412
nfrac1413
nfrac1414
nfrac1415
Critical
value(A
)P(A
)gtfrac14A
Critical
value(A
)P(A
)gtfrac14A
Critical
value(A
)P(A
)gtfrac14A
Critical
value(A
)P(A
)gtfrac14A
17
02015
19
02056
21
02168
23
02125
18
01677
20
01733
22
01909
24
01873
19
01407
21
01488
23
01657
25
01648
20
01162
22
01249
24
01431
26
01454
21
00929
23
01038
25
01267
27
01285
22
00755
24
00878
26
01100
28
01132
23
00615
25
00736
27
00945
29
00972
24
00507
26
00606
28
00806
30
00841
25
00416
27
00484
29
00691
31
00723
26
00315
28
00410
30
00581
32
00624
27
00240
29
00334
31
00497
33
00542
28
00175
30
00278
32
00435
34
00468
29
00134
31
00221
33
00360
35
00403
30
00099
32
00184
34
00293
36
00344
33
00145
35
00236
37
00296
34
00120
36
00200
38
00240
35
00097
37
00175
39
00206
38
00141
40
00163
39
00119
41
00141
40
00085
42
00118
43
00089
A New One-Sample Test for Goodness-of-Fit 191
Dow
nloa
ded
by [
Uni
vers
ity o
f C
alif
orni
a R
iver
side
Lib
rari
es]
at 1
005
03
Nov
embe
r 20
14
ORDER REPRINTS
(Table 7) The alternative distribution functions are described in sectiontwo of this paper The sample sizes (n) were set equal to 10 20 and 40The alpha level was set at 010
REFERENCES
Bradley J V (1968) Distribution-Free Statistical Tests EnglewoodCliffs NJ Prentice-Hall
Gibbons J D (1976) Nonparametric Methods for Quantitative AnalysisNew York Holt Rinehart and Winston
Gibbons J D (1985) Nonparametric Statistical Inference New YorkMarcel Dekker
Table 7
Alternative D W2 V U2 A2 Q Chi2 A
(nfrac14 10 alphafrac14 010)Fkfrac1415 023 027 018 019 024 043 ndash 027Fkfrac1420 054 060 035 035 058 ndash ndash 058Gkfrac1415 009 007 022 023 006 ndash ndash 007Gkfrac1420 009 007 040 044 006 ndash ndash 010Gkfrac1430 021 021 081 086 018 ndash ndash 029Hkfrac1415 ndash ndash ndash ndash ndash ndash ndash 014Hkfrac1420 ndash ndash ndash ndash ndash ndash ndash 021
(nfrac14 20 alphafrac14 010)Fkfrac1415 038 046 025 028 046 068 ndash 046Fkfrac1420 078 087 061 060 087 097 059 088Gkfrac1415 013 011 032 034 010 011 ndash 010Gkfrac1420 025 025 071 077 028 025 ndash 029Gkfrac1430 063 079 099 099 084 ndash ndash 083Hkfrac1415 025 020 036 037 028 ndash ndash 017Hkfrac1420 047 044 071 077 054 ndash ndash 036
(nfrac14 40 alphafrac14 010)Fkfrac1415 060 070 043 043 ndash 089 040 073Fkfrac1420 098 099 091 089 ndash 100 089 099Gkfrac1415 019 022 057 061 ndash ndash 039 023Gkfrac1420 056 072 096 098 ndash ndash 085 075Gkfrac1430 ndash ndash ndash ndash ndash ndash ndash 100Hkfrac1415 036 032 058 063 ndash ndash ndash 027Hkfrac1420 071 080 096 098 ndash ndash ndash 075
192 Damico
Dow
nloa
ded
by [
Uni
vers
ity o
f C
alif
orni
a R
iver
side
Lib
rari
es]
at 1
005
03
Nov
embe
r 20
14
ORDER REPRINTS
Hollander M Wolfe D A (1973) Nonparametric Statistical Methods2nd ed New York Wiley
Stephens M A (1974) EDF statistics for goodness-of-fit and somecomparisons (In theory and methods) J Amer Stat Assoc Theoryand Methods Section September 69(347)730ndash737
A New One-Sample Test for Goodness-of-Fit 193
Dow
nloa
ded
by [
Uni
vers
ity o
f C
alif
orni
a R
iver
side
Lib
rari
es]
at 1
005
03
Nov
embe
r 20
14
Request PermissionOrder Reprints
Reprints of this article can also be ordered at
httpwwwdekkercomservletproductDOI101081STA120026585
Request Permission or Order Reprints Instantly
Interested in copying and sharing this article In most cases US Copyright Law requires that you get permission from the articlersquos rightsholder before using copyrighted content
All information and materials found in this article including but not limited to text trademarks patents logos graphics and images (the Materials) are the copyrighted works and other forms of intellectual property of Marcel Dekker Inc or its licensors All rights not expressly granted are reserved
Get permission to lawfully reproduce and distribute the Materials or order reprints quickly and painlessly Simply click on the Request Permission Order Reprints link below and follow the instructions Visit the US Copyright Office for information on Fair Use limitations of US copyright law Please refer to The Association of American Publishersrsquo (AAP) website for guidelines on Fair Use in the Classroom
The Materials are for your personal use only and cannot be reformatted reposted resold or distributed by electronic means or otherwise without permission from Marcel Dekker Inc Marcel Dekker Inc grants you the limited right to display the Materials only on your personal computer or personal wireless device and to copy and download single copies of such Materials provided that any copyright trademark or other notice appearing on such Materials is also retained by displayed copied or downloaded as part of the Materials and is not removed or obscured and provided you do not edit modify alter or enhance the Materials Please refer to our Website User Agreement for more details
Dow
nloa
ded
by [
Uni
vers
ity o
f C
alif
orni
a R
iver
side
Lib
rari
es]
at 1
005
03
Nov
embe
r 20
14
ORDER REPRINTS
Table 4 Continued
N nCritical
value (A) P(A) gtfrac14A N nCritical
value (A) P(A) gtfrac14A
1600 28 59 01981 1600 38 96 02019
73 01019 118 0101386 00500 135 00500110 00100 181 00100
1600 29 66 01975 1600 39 99 0201380 01013 120 0100092 00481 140 00494117 00100 170 00094
10000 30 67 02029 10000 40 103 0201181 01025 126 0100095 00490 148 00496126 00095 191 00099
1600 31 71 01969 10000 50 145 0194587 00981 177 00984101 00500 205 00496128 00100 265 00099
1600 32 74 02038 10000 60 190 0199991 01000 232 00986107 00488 270 00498135 00100 348 00093
1600 33 77 01963 10000 70 240 0201990 01006 292 01027108 00500 340 00488141 00094 438 00095
1600 34 82 01975 10000 80 293 01988101 00988 357 01015117 00488 415 00530149 00100 535 00112
10000 35 84 02036 10000 90 349 01982102 01019 426 01015120 00493 495 00530152 00103 639 00112
1600 36 91 02019 10000 100 409 01930113 00981 499 00955132 00488 580 00467166 00100 748 00089
1600 37 90 01988110 01000128 00488172 00100
A New One-Sample Test for Goodness-of-Fit 189
Dow
nloa
ded
by [
Uni
vers
ity o
f C
alif
orni
a R
iver
side
Lib
rari
es]
at 1
005
03
Nov
embe
r 20
14
ORDER REPRINTS
Table
5
nfrac148
nfrac149
nfrac1410
nfrac1411
Critical
value(A
)P(A
)gtfrac14A
Critical
value(A
)P(A
)gtfrac14A
Critical
value(A
)P(A
)gtfrac14A
Critical
value(A
)P(A
)gtfrac14A
902267
11
02074
13
02008
15
02018
10
01657
12
01580
14
01606
16
01628
11
01183
13
01176
15
01260
17
01336
12
00810
14
00874
16
00972
18
01091
13
00543
15
00633
17
00729
19
00860
14
00347
16
00462
18
00542
20
00670
15
00227
17
00335
19
00406
21
00522
16
00151
18
00241
20
00298
22
00417
17
00089
19
00157
21
00221
23
00334
20
00106
22
00166
24
00257
21
00062
23
00126
25
00194
24
00090
26
00147
27
00111
28
00082
190 Damico
Dow
nloa
ded
by [
Uni
vers
ity o
f C
alif
orni
a R
iver
side
Lib
rari
es]
at 1
005
03
Nov
embe
r 20
14
ORDER REPRINTS
Table
6
nfrac1412
nfrac1413
nfrac1414
nfrac1415
Critical
value(A
)P(A
)gtfrac14A
Critical
value(A
)P(A
)gtfrac14A
Critical
value(A
)P(A
)gtfrac14A
Critical
value(A
)P(A
)gtfrac14A
17
02015
19
02056
21
02168
23
02125
18
01677
20
01733
22
01909
24
01873
19
01407
21
01488
23
01657
25
01648
20
01162
22
01249
24
01431
26
01454
21
00929
23
01038
25
01267
27
01285
22
00755
24
00878
26
01100
28
01132
23
00615
25
00736
27
00945
29
00972
24
00507
26
00606
28
00806
30
00841
25
00416
27
00484
29
00691
31
00723
26
00315
28
00410
30
00581
32
00624
27
00240
29
00334
31
00497
33
00542
28
00175
30
00278
32
00435
34
00468
29
00134
31
00221
33
00360
35
00403
30
00099
32
00184
34
00293
36
00344
33
00145
35
00236
37
00296
34
00120
36
00200
38
00240
35
00097
37
00175
39
00206
38
00141
40
00163
39
00119
41
00141
40
00085
42
00118
43
00089
A New One-Sample Test for Goodness-of-Fit 191
Dow
nloa
ded
by [
Uni
vers
ity o
f C
alif
orni
a R
iver
side
Lib
rari
es]
at 1
005
03
Nov
embe
r 20
14
ORDER REPRINTS
(Table 7) The alternative distribution functions are described in sectiontwo of this paper The sample sizes (n) were set equal to 10 20 and 40The alpha level was set at 010
REFERENCES
Bradley J V (1968) Distribution-Free Statistical Tests EnglewoodCliffs NJ Prentice-Hall
Gibbons J D (1976) Nonparametric Methods for Quantitative AnalysisNew York Holt Rinehart and Winston
Gibbons J D (1985) Nonparametric Statistical Inference New YorkMarcel Dekker
Table 7
Alternative D W2 V U2 A2 Q Chi2 A
(nfrac14 10 alphafrac14 010)Fkfrac1415 023 027 018 019 024 043 ndash 027Fkfrac1420 054 060 035 035 058 ndash ndash 058Gkfrac1415 009 007 022 023 006 ndash ndash 007Gkfrac1420 009 007 040 044 006 ndash ndash 010Gkfrac1430 021 021 081 086 018 ndash ndash 029Hkfrac1415 ndash ndash ndash ndash ndash ndash ndash 014Hkfrac1420 ndash ndash ndash ndash ndash ndash ndash 021
(nfrac14 20 alphafrac14 010)Fkfrac1415 038 046 025 028 046 068 ndash 046Fkfrac1420 078 087 061 060 087 097 059 088Gkfrac1415 013 011 032 034 010 011 ndash 010Gkfrac1420 025 025 071 077 028 025 ndash 029Gkfrac1430 063 079 099 099 084 ndash ndash 083Hkfrac1415 025 020 036 037 028 ndash ndash 017Hkfrac1420 047 044 071 077 054 ndash ndash 036
(nfrac14 40 alphafrac14 010)Fkfrac1415 060 070 043 043 ndash 089 040 073Fkfrac1420 098 099 091 089 ndash 100 089 099Gkfrac1415 019 022 057 061 ndash ndash 039 023Gkfrac1420 056 072 096 098 ndash ndash 085 075Gkfrac1430 ndash ndash ndash ndash ndash ndash ndash 100Hkfrac1415 036 032 058 063 ndash ndash ndash 027Hkfrac1420 071 080 096 098 ndash ndash ndash 075
192 Damico
Dow
nloa
ded
by [
Uni
vers
ity o
f C
alif
orni
a R
iver
side
Lib
rari
es]
at 1
005
03
Nov
embe
r 20
14
ORDER REPRINTS
Hollander M Wolfe D A (1973) Nonparametric Statistical Methods2nd ed New York Wiley
Stephens M A (1974) EDF statistics for goodness-of-fit and somecomparisons (In theory and methods) J Amer Stat Assoc Theoryand Methods Section September 69(347)730ndash737
A New One-Sample Test for Goodness-of-Fit 193
Dow
nloa
ded
by [
Uni
vers
ity o
f C
alif
orni
a R
iver
side
Lib
rari
es]
at 1
005
03
Nov
embe
r 20
14
Request PermissionOrder Reprints
Reprints of this article can also be ordered at
httpwwwdekkercomservletproductDOI101081STA120026585
Request Permission or Order Reprints Instantly
Interested in copying and sharing this article In most cases US Copyright Law requires that you get permission from the articlersquos rightsholder before using copyrighted content
All information and materials found in this article including but not limited to text trademarks patents logos graphics and images (the Materials) are the copyrighted works and other forms of intellectual property of Marcel Dekker Inc or its licensors All rights not expressly granted are reserved
Get permission to lawfully reproduce and distribute the Materials or order reprints quickly and painlessly Simply click on the Request Permission Order Reprints link below and follow the instructions Visit the US Copyright Office for information on Fair Use limitations of US copyright law Please refer to The Association of American Publishersrsquo (AAP) website for guidelines on Fair Use in the Classroom
The Materials are for your personal use only and cannot be reformatted reposted resold or distributed by electronic means or otherwise without permission from Marcel Dekker Inc Marcel Dekker Inc grants you the limited right to display the Materials only on your personal computer or personal wireless device and to copy and download single copies of such Materials provided that any copyright trademark or other notice appearing on such Materials is also retained by displayed copied or downloaded as part of the Materials and is not removed or obscured and provided you do not edit modify alter or enhance the Materials Please refer to our Website User Agreement for more details
Dow
nloa
ded
by [
Uni
vers
ity o
f C
alif
orni
a R
iver
side
Lib
rari
es]
at 1
005
03
Nov
embe
r 20
14
ORDER REPRINTS
Table
5
nfrac148
nfrac149
nfrac1410
nfrac1411
Critical
value(A
)P(A
)gtfrac14A
Critical
value(A
)P(A
)gtfrac14A
Critical
value(A
)P(A
)gtfrac14A
Critical
value(A
)P(A
)gtfrac14A
902267
11
02074
13
02008
15
02018
10
01657
12
01580
14
01606
16
01628
11
01183
13
01176
15
01260
17
01336
12
00810
14
00874
16
00972
18
01091
13
00543
15
00633
17
00729
19
00860
14
00347
16
00462
18
00542
20
00670
15
00227
17
00335
19
00406
21
00522
16
00151
18
00241
20
00298
22
00417
17
00089
19
00157
21
00221
23
00334
20
00106
22
00166
24
00257
21
00062
23
00126
25
00194
24
00090
26
00147
27
00111
28
00082
190 Damico
Dow
nloa
ded
by [
Uni
vers
ity o
f C
alif
orni
a R
iver
side
Lib
rari
es]
at 1
005
03
Nov
embe
r 20
14
ORDER REPRINTS
Table
6
nfrac1412
nfrac1413
nfrac1414
nfrac1415
Critical
value(A
)P(A
)gtfrac14A
Critical
value(A
)P(A
)gtfrac14A
Critical
value(A
)P(A
)gtfrac14A
Critical
value(A
)P(A
)gtfrac14A
17
02015
19
02056
21
02168
23
02125
18
01677
20
01733
22
01909
24
01873
19
01407
21
01488
23
01657
25
01648
20
01162
22
01249
24
01431
26
01454
21
00929
23
01038
25
01267
27
01285
22
00755
24
00878
26
01100
28
01132
23
00615
25
00736
27
00945
29
00972
24
00507
26
00606
28
00806
30
00841
25
00416
27
00484
29
00691
31
00723
26
00315
28
00410
30
00581
32
00624
27
00240
29
00334
31
00497
33
00542
28
00175
30
00278
32
00435
34
00468
29
00134
31
00221
33
00360
35
00403
30
00099
32
00184
34
00293
36
00344
33
00145
35
00236
37
00296
34
00120
36
00200
38
00240
35
00097
37
00175
39
00206
38
00141
40
00163
39
00119
41
00141
40
00085
42
00118
43
00089
A New One-Sample Test for Goodness-of-Fit 191
Dow
nloa
ded
by [
Uni
vers
ity o
f C
alif
orni
a R
iver
side
Lib
rari
es]
at 1
005
03
Nov
embe
r 20
14
ORDER REPRINTS
(Table 7) The alternative distribution functions are described in sectiontwo of this paper The sample sizes (n) were set equal to 10 20 and 40The alpha level was set at 010
REFERENCES
Bradley J V (1968) Distribution-Free Statistical Tests EnglewoodCliffs NJ Prentice-Hall
Gibbons J D (1976) Nonparametric Methods for Quantitative AnalysisNew York Holt Rinehart and Winston
Gibbons J D (1985) Nonparametric Statistical Inference New YorkMarcel Dekker
Table 7
Alternative D W2 V U2 A2 Q Chi2 A
(nfrac14 10 alphafrac14 010)Fkfrac1415 023 027 018 019 024 043 ndash 027Fkfrac1420 054 060 035 035 058 ndash ndash 058Gkfrac1415 009 007 022 023 006 ndash ndash 007Gkfrac1420 009 007 040 044 006 ndash ndash 010Gkfrac1430 021 021 081 086 018 ndash ndash 029Hkfrac1415 ndash ndash ndash ndash ndash ndash ndash 014Hkfrac1420 ndash ndash ndash ndash ndash ndash ndash 021
(nfrac14 20 alphafrac14 010)Fkfrac1415 038 046 025 028 046 068 ndash 046Fkfrac1420 078 087 061 060 087 097 059 088Gkfrac1415 013 011 032 034 010 011 ndash 010Gkfrac1420 025 025 071 077 028 025 ndash 029Gkfrac1430 063 079 099 099 084 ndash ndash 083Hkfrac1415 025 020 036 037 028 ndash ndash 017Hkfrac1420 047 044 071 077 054 ndash ndash 036
(nfrac14 40 alphafrac14 010)Fkfrac1415 060 070 043 043 ndash 089 040 073Fkfrac1420 098 099 091 089 ndash 100 089 099Gkfrac1415 019 022 057 061 ndash ndash 039 023Gkfrac1420 056 072 096 098 ndash ndash 085 075Gkfrac1430 ndash ndash ndash ndash ndash ndash ndash 100Hkfrac1415 036 032 058 063 ndash ndash ndash 027Hkfrac1420 071 080 096 098 ndash ndash ndash 075
192 Damico
Dow
nloa
ded
by [
Uni
vers
ity o
f C
alif
orni
a R
iver
side
Lib
rari
es]
at 1
005
03
Nov
embe
r 20
14
ORDER REPRINTS
Hollander M Wolfe D A (1973) Nonparametric Statistical Methods2nd ed New York Wiley
Stephens M A (1974) EDF statistics for goodness-of-fit and somecomparisons (In theory and methods) J Amer Stat Assoc Theoryand Methods Section September 69(347)730ndash737
A New One-Sample Test for Goodness-of-Fit 193
Dow
nloa
ded
by [
Uni
vers
ity o
f C
alif
orni
a R
iver
side
Lib
rari
es]
at 1
005
03
Nov
embe
r 20
14
Request PermissionOrder Reprints
Reprints of this article can also be ordered at
httpwwwdekkercomservletproductDOI101081STA120026585
Request Permission or Order Reprints Instantly
Interested in copying and sharing this article In most cases US Copyright Law requires that you get permission from the articlersquos rightsholder before using copyrighted content
All information and materials found in this article including but not limited to text trademarks patents logos graphics and images (the Materials) are the copyrighted works and other forms of intellectual property of Marcel Dekker Inc or its licensors All rights not expressly granted are reserved
Get permission to lawfully reproduce and distribute the Materials or order reprints quickly and painlessly Simply click on the Request Permission Order Reprints link below and follow the instructions Visit the US Copyright Office for information on Fair Use limitations of US copyright law Please refer to The Association of American Publishersrsquo (AAP) website for guidelines on Fair Use in the Classroom
The Materials are for your personal use only and cannot be reformatted reposted resold or distributed by electronic means or otherwise without permission from Marcel Dekker Inc Marcel Dekker Inc grants you the limited right to display the Materials only on your personal computer or personal wireless device and to copy and download single copies of such Materials provided that any copyright trademark or other notice appearing on such Materials is also retained by displayed copied or downloaded as part of the Materials and is not removed or obscured and provided you do not edit modify alter or enhance the Materials Please refer to our Website User Agreement for more details
Dow
nloa
ded
by [
Uni
vers
ity o
f C
alif
orni
a R
iver
side
Lib
rari
es]
at 1
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embe
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ORDER REPRINTS
Table
6
nfrac1412
nfrac1413
nfrac1414
nfrac1415
Critical
value(A
)P(A
)gtfrac14A
Critical
value(A
)P(A
)gtfrac14A
Critical
value(A
)P(A
)gtfrac14A
Critical
value(A
)P(A
)gtfrac14A
17
02015
19
02056
21
02168
23
02125
18
01677
20
01733
22
01909
24
01873
19
01407
21
01488
23
01657
25
01648
20
01162
22
01249
24
01431
26
01454
21
00929
23
01038
25
01267
27
01285
22
00755
24
00878
26
01100
28
01132
23
00615
25
00736
27
00945
29
00972
24
00507
26
00606
28
00806
30
00841
25
00416
27
00484
29
00691
31
00723
26
00315
28
00410
30
00581
32
00624
27
00240
29
00334
31
00497
33
00542
28
00175
30
00278
32
00435
34
00468
29
00134
31
00221
33
00360
35
00403
30
00099
32
00184
34
00293
36
00344
33
00145
35
00236
37
00296
34
00120
36
00200
38
00240
35
00097
37
00175
39
00206
38
00141
40
00163
39
00119
41
00141
40
00085
42
00118
43
00089
A New One-Sample Test for Goodness-of-Fit 191
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(Table 7) The alternative distribution functions are described in sectiontwo of this paper The sample sizes (n) were set equal to 10 20 and 40The alpha level was set at 010
REFERENCES
Bradley J V (1968) Distribution-Free Statistical Tests EnglewoodCliffs NJ Prentice-Hall
Gibbons J D (1976) Nonparametric Methods for Quantitative AnalysisNew York Holt Rinehart and Winston
Gibbons J D (1985) Nonparametric Statistical Inference New YorkMarcel Dekker
Table 7
Alternative D W2 V U2 A2 Q Chi2 A
(nfrac14 10 alphafrac14 010)Fkfrac1415 023 027 018 019 024 043 ndash 027Fkfrac1420 054 060 035 035 058 ndash ndash 058Gkfrac1415 009 007 022 023 006 ndash ndash 007Gkfrac1420 009 007 040 044 006 ndash ndash 010Gkfrac1430 021 021 081 086 018 ndash ndash 029Hkfrac1415 ndash ndash ndash ndash ndash ndash ndash 014Hkfrac1420 ndash ndash ndash ndash ndash ndash ndash 021
(nfrac14 20 alphafrac14 010)Fkfrac1415 038 046 025 028 046 068 ndash 046Fkfrac1420 078 087 061 060 087 097 059 088Gkfrac1415 013 011 032 034 010 011 ndash 010Gkfrac1420 025 025 071 077 028 025 ndash 029Gkfrac1430 063 079 099 099 084 ndash ndash 083Hkfrac1415 025 020 036 037 028 ndash ndash 017Hkfrac1420 047 044 071 077 054 ndash ndash 036
(nfrac14 40 alphafrac14 010)Fkfrac1415 060 070 043 043 ndash 089 040 073Fkfrac1420 098 099 091 089 ndash 100 089 099Gkfrac1415 019 022 057 061 ndash ndash 039 023Gkfrac1420 056 072 096 098 ndash ndash 085 075Gkfrac1430 ndash ndash ndash ndash ndash ndash ndash 100Hkfrac1415 036 032 058 063 ndash ndash ndash 027Hkfrac1420 071 080 096 098 ndash ndash ndash 075
192 Damico
Dow
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a R
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side
Lib
rari
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005
03
Nov
embe
r 20
14
ORDER REPRINTS
Hollander M Wolfe D A (1973) Nonparametric Statistical Methods2nd ed New York Wiley
Stephens M A (1974) EDF statistics for goodness-of-fit and somecomparisons (In theory and methods) J Amer Stat Assoc Theoryand Methods Section September 69(347)730ndash737
A New One-Sample Test for Goodness-of-Fit 193
Dow
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by [
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ity o
f C
alif
orni
a R
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side
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rari
es]
at 1
005
03
Nov
embe
r 20
14
Request PermissionOrder Reprints
Reprints of this article can also be ordered at
httpwwwdekkercomservletproductDOI101081STA120026585
Request Permission or Order Reprints Instantly
Interested in copying and sharing this article In most cases US Copyright Law requires that you get permission from the articlersquos rightsholder before using copyrighted content
All information and materials found in this article including but not limited to text trademarks patents logos graphics and images (the Materials) are the copyrighted works and other forms of intellectual property of Marcel Dekker Inc or its licensors All rights not expressly granted are reserved
Get permission to lawfully reproduce and distribute the Materials or order reprints quickly and painlessly Simply click on the Request Permission Order Reprints link below and follow the instructions Visit the US Copyright Office for information on Fair Use limitations of US copyright law Please refer to The Association of American Publishersrsquo (AAP) website for guidelines on Fair Use in the Classroom
The Materials are for your personal use only and cannot be reformatted reposted resold or distributed by electronic means or otherwise without permission from Marcel Dekker Inc Marcel Dekker Inc grants you the limited right to display the Materials only on your personal computer or personal wireless device and to copy and download single copies of such Materials provided that any copyright trademark or other notice appearing on such Materials is also retained by displayed copied or downloaded as part of the Materials and is not removed or obscured and provided you do not edit modify alter or enhance the Materials Please refer to our Website User Agreement for more details
Dow
nloa
ded
by [
Uni
vers
ity o
f C
alif
orni
a R
iver
side
Lib
rari
es]
at 1
005
03
Nov
embe
r 20
14
ORDER REPRINTS
(Table 7) The alternative distribution functions are described in sectiontwo of this paper The sample sizes (n) were set equal to 10 20 and 40The alpha level was set at 010
REFERENCES
Bradley J V (1968) Distribution-Free Statistical Tests EnglewoodCliffs NJ Prentice-Hall
Gibbons J D (1976) Nonparametric Methods for Quantitative AnalysisNew York Holt Rinehart and Winston
Gibbons J D (1985) Nonparametric Statistical Inference New YorkMarcel Dekker
Table 7
Alternative D W2 V U2 A2 Q Chi2 A
(nfrac14 10 alphafrac14 010)Fkfrac1415 023 027 018 019 024 043 ndash 027Fkfrac1420 054 060 035 035 058 ndash ndash 058Gkfrac1415 009 007 022 023 006 ndash ndash 007Gkfrac1420 009 007 040 044 006 ndash ndash 010Gkfrac1430 021 021 081 086 018 ndash ndash 029Hkfrac1415 ndash ndash ndash ndash ndash ndash ndash 014Hkfrac1420 ndash ndash ndash ndash ndash ndash ndash 021
(nfrac14 20 alphafrac14 010)Fkfrac1415 038 046 025 028 046 068 ndash 046Fkfrac1420 078 087 061 060 087 097 059 088Gkfrac1415 013 011 032 034 010 011 ndash 010Gkfrac1420 025 025 071 077 028 025 ndash 029Gkfrac1430 063 079 099 099 084 ndash ndash 083Hkfrac1415 025 020 036 037 028 ndash ndash 017Hkfrac1420 047 044 071 077 054 ndash ndash 036
(nfrac14 40 alphafrac14 010)Fkfrac1415 060 070 043 043 ndash 089 040 073Fkfrac1420 098 099 091 089 ndash 100 089 099Gkfrac1415 019 022 057 061 ndash ndash 039 023Gkfrac1420 056 072 096 098 ndash ndash 085 075Gkfrac1430 ndash ndash ndash ndash ndash ndash ndash 100Hkfrac1415 036 032 058 063 ndash ndash ndash 027Hkfrac1420 071 080 096 098 ndash ndash ndash 075
192 Damico
Dow
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by [
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f C
alif
orni
a R
iver
side
Lib
rari
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at 1
005
03
Nov
embe
r 20
14
ORDER REPRINTS
Hollander M Wolfe D A (1973) Nonparametric Statistical Methods2nd ed New York Wiley
Stephens M A (1974) EDF statistics for goodness-of-fit and somecomparisons (In theory and methods) J Amer Stat Assoc Theoryand Methods Section September 69(347)730ndash737
A New One-Sample Test for Goodness-of-Fit 193
Dow
nloa
ded
by [
Uni
vers
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Request PermissionOrder Reprints
Reprints of this article can also be ordered at
httpwwwdekkercomservletproductDOI101081STA120026585
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Interested in copying and sharing this article In most cases US Copyright Law requires that you get permission from the articlersquos rightsholder before using copyrighted content
All information and materials found in this article including but not limited to text trademarks patents logos graphics and images (the Materials) are the copyrighted works and other forms of intellectual property of Marcel Dekker Inc or its licensors All rights not expressly granted are reserved
Get permission to lawfully reproduce and distribute the Materials or order reprints quickly and painlessly Simply click on the Request Permission Order Reprints link below and follow the instructions Visit the US Copyright Office for information on Fair Use limitations of US copyright law Please refer to The Association of American Publishersrsquo (AAP) website for guidelines on Fair Use in the Classroom
The Materials are for your personal use only and cannot be reformatted reposted resold or distributed by electronic means or otherwise without permission from Marcel Dekker Inc Marcel Dekker Inc grants you the limited right to display the Materials only on your personal computer or personal wireless device and to copy and download single copies of such Materials provided that any copyright trademark or other notice appearing on such Materials is also retained by displayed copied or downloaded as part of the Materials and is not removed or obscured and provided you do not edit modify alter or enhance the Materials Please refer to our Website User Agreement for more details
Dow
nloa
ded
by [
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f C
alif
orni
a R
iver
side
Lib
rari
es]
at 1
005
03
Nov
embe
r 20
14
ORDER REPRINTS
Hollander M Wolfe D A (1973) Nonparametric Statistical Methods2nd ed New York Wiley
Stephens M A (1974) EDF statistics for goodness-of-fit and somecomparisons (In theory and methods) J Amer Stat Assoc Theoryand Methods Section September 69(347)730ndash737
A New One-Sample Test for Goodness-of-Fit 193
Dow
nloa
ded
by [
Uni
vers
ity o
f C
alif
orni
a R
iver
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rari
es]
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Request PermissionOrder Reprints
Reprints of this article can also be ordered at
httpwwwdekkercomservletproductDOI101081STA120026585
Request Permission or Order Reprints Instantly
Interested in copying and sharing this article In most cases US Copyright Law requires that you get permission from the articlersquos rightsholder before using copyrighted content
All information and materials found in this article including but not limited to text trademarks patents logos graphics and images (the Materials) are the copyrighted works and other forms of intellectual property of Marcel Dekker Inc or its licensors All rights not expressly granted are reserved
Get permission to lawfully reproduce and distribute the Materials or order reprints quickly and painlessly Simply click on the Request Permission Order Reprints link below and follow the instructions Visit the US Copyright Office for information on Fair Use limitations of US copyright law Please refer to The Association of American Publishersrsquo (AAP) website for guidelines on Fair Use in the Classroom
The Materials are for your personal use only and cannot be reformatted reposted resold or distributed by electronic means or otherwise without permission from Marcel Dekker Inc Marcel Dekker Inc grants you the limited right to display the Materials only on your personal computer or personal wireless device and to copy and download single copies of such Materials provided that any copyright trademark or other notice appearing on such Materials is also retained by displayed copied or downloaded as part of the Materials and is not removed or obscured and provided you do not edit modify alter or enhance the Materials Please refer to our Website User Agreement for more details
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Request PermissionOrder Reprints
Reprints of this article can also be ordered at
httpwwwdekkercomservletproductDOI101081STA120026585
Request Permission or Order Reprints Instantly
Interested in copying and sharing this article In most cases US Copyright Law requires that you get permission from the articlersquos rightsholder before using copyrighted content
All information and materials found in this article including but not limited to text trademarks patents logos graphics and images (the Materials) are the copyrighted works and other forms of intellectual property of Marcel Dekker Inc or its licensors All rights not expressly granted are reserved
Get permission to lawfully reproduce and distribute the Materials or order reprints quickly and painlessly Simply click on the Request Permission Order Reprints link below and follow the instructions Visit the US Copyright Office for information on Fair Use limitations of US copyright law Please refer to The Association of American Publishersrsquo (AAP) website for guidelines on Fair Use in the Classroom
The Materials are for your personal use only and cannot be reformatted reposted resold or distributed by electronic means or otherwise without permission from Marcel Dekker Inc Marcel Dekker Inc grants you the limited right to display the Materials only on your personal computer or personal wireless device and to copy and download single copies of such Materials provided that any copyright trademark or other notice appearing on such Materials is also retained by displayed copied or downloaded as part of the Materials and is not removed or obscured and provided you do not edit modify alter or enhance the Materials Please refer to our Website User Agreement for more details
Dow
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ded
by [
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a R
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side
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at 1
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embe
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